Method and system for modeling bone structure

ABSTRACT

The present invention discloses a structural and mechanical model and modeling methods for human bone based on bone&#39;s hierarchical structure and on its hierarchical mechanical behavior. The model allows for the assessment of bone deformations, computation of strains and stresses due to the specific forces acting on bone during function, and contemplates forces that do or do not cause viscous effects and forces that cause either elastic or plastic bone deformation.

This application is a continuation-in-part of U.S. patent applicationSer. No. 10/429,491 (now U.S. Pat. No. 7,212,958), filed May 5, 2003,which is a continuation-in-part of U.S. patent application Ser. No.10/357,090 (now U.S. Pat. No. 7,283,940), filed Feb. 3, 2003, whichclaims priority under 35 U.S.C. §119 from U.S. Provisional PatentApplication Ser. No. 60/353,768, filed Feb. 1, 2002, and U.S.Provisional Patent Application Ser. No. 60/380,174, filed May 6, 2002;each of which is hereby incorporated by reference in its entirety; andwhich is also a continuation-in-part of U.S. patent application Ser. No.10/066,293 (now U.S. Pat. No. 7,127,383), filed Jan. 31, 2002, which isa continuation in part of U.S. patent application Ser. No. 09/981,684(now U.S. Pat. No. 7,124,067), filed Oct. 17, 2001. Each of theseapplications is hereby incorporated by reference in its entirety.

FIELD OF THE INVENTION

The present invention discloses a structural and mechanical model andmodeling methods for human bone based on bone's hierarchical structureand on its hierarchical mechanical behavior. The model allows for theassessment of bone deformations and the computation of strains andstresses due to the specific forces acting on bone during function. Themodel further contemplates forces that do or do not cause viscouseffects and forces that cause either elastic or plastic bonedeformations characterized by fractures. In preferred embodiments themodel is computerized, for example using computer simulation, imagingand rendering techniques.

This invention also relates to the viscoelastic behavior of humancompact bone's hierarchical structure. Specifically, the presentinvention relates to modeling of viscoelastic osteon behavior undertorsional loading. This invention further relates to morphological andmechanical properties of isolated osteonic lamellae, which represent themicrostructural building blocks of individual secodnary osteons andhence of adult human compact bone tissue. Morphology and mechanicalfunction is related to the properties of the collagen bundle orientationand density distribution, and the behavior under quasi-static tensionalloading such bone behavior is also incorporated into the computerizedmodel.

BACKGROUND OF THE INVENTION

Material science defines the structural properties of an object as theproperties that describe the object's makeup independent from its shape.Adult human bone has a complex structure and can be described as afour-order hierarchy, arranged in decreasing size (Petersen, 1930). Thefirst order, macrostructure (FIG. 1), comprises the structurescorresponding to gross shape and differentiation between compact (orcortical) bone (FIG. 2) and spongy (cancellous or trabecular) bone (FIG.3). Compact bone is present in the long bone shaft (or diaphysis).Spongy bone is present in the lower jaw (mandible), in the epiphysis oflong bone shaft, and in flat and short bones. The second order (ormicrostructure) of compact bone includes lamellar systems (lamellae).Organized lamellae around vascular canals are referred to as osteons(harvesian systems) and disorganized lamellae among osteons are referredto as the interstitial bone. The second order also is comprised ofrelated structures such as bone marrow (see e.g. Bloom and Fawcetts,1986). The third order (or ultrastructure) of compact bone consistsmainly of collagen bundles and hydroxyapatite crystallites;mucopolysaccarides amount to a small amout but may have a significantrole. The fourth order of compact bone consists of moleculararrangements between organic and inorganic substances. For cancellousbone, the second order includes trabeculae, which comprise lamellarsystems and related structures, e.g. bone marrow. The third and forthorders of cancellous bone are the same those described for compact bone.

The osteon comprises a haversian canal with concentrically arrangedlamellae. Osteons of long bone are generally directed along the longbone axis. Osteonic lamellae are organized as consisting of an organicframework (mostly a collagen bundle) embedded in ground substances, suchas proteins and water, and hydroxyapatite crystallites. Thehydroxyapatite crystallites are oriented in directions analogous tothose of the bundles. Osteons measure a few centimeters in length andare between 200 and 300 μm in diameter. The degree of osteoncalcification (relative amount of hydroxyapatite crystallites) isvariable from osteon to osteon as well as within osteons. Thesedifferences are proposed to be due to the process of bone renewal orremodeling. In this process, osteons are renewed continuously.Consequently, osteons at different degrees of calcification are alwayspresent in adult compact bone.

There is a spectrum of osteon types that refer to the arrangements offiber bundle direction in the lamellae. Two osteon types, “longitudinal”and “alternate”, are representative of the two ends of the spectrum.Longitudinal osteons consist of bundles with a marked longitudinalspiral course. Alternate osteons consist of bundles with a markedlongitudinal, oblique, and transversal course in successive lamellae(Frasca et al., 1977, Giraud-Guille, 1988, Ascenzi M.-G. et al., 2003).There are two types of lamellae, termed extinct (or longitudinal) andbright (or transverse or circularly-fibered) lamellae. Extinct (or dark)lamellae appear extinct whereas bright lamellae appear bright under apolarizing microscope when the microscope and osteon axes are aligned.

Compact bone consists of about 40% minerals, 40% collagen, and 20%fluids. The major internal spaces or discontinuities of compact boneinclude the vascular system, pits and cavities (lacunae), narrowchannels (canaliculae), fine porosity, and spaces between the mineralphases. The major internal material discontinuities of compact bone(FIG. 5), in order of decreasing size, are:

Vascular system 20-50 μm Lacunae 4-6 μm Canaliculae 0.5-2 μm Fineporosity 600-800 Å Spaces between mineral phases 50-100 Å

Cancellous bone consists of trabeculae, i.e. osseous structures witheither a sheet-like or a rod-like configuration. These structuresinterlace to form a lattice-like or spongy biological structure (FIG.3). For example, both types of trabeculae are present in the calcaneous;however, up to 3% of the rod-like configurations are tubular due to thevascular canal running through them. Therefore, they are similar to theharvesian system. In general, tubular trabeculae appear to have arelatively simple structure. Collagen fibrils run mostly parallel to thelong axis of tubular trabeculae in the trabeculae outer portion andperpendicular in the inner portion. Although the true density of fullycalcified cancellous bone is a little lower and the proteoglycan contenta little greater than those of the fully calcified compact bone, thesubstantial difference between compact and cancellous bone resides inthe porosity. The cancellous bone porosity, which ranges from 30% tomore than 90%, is mainly due to the wide vascular and bone marrowintrabecular spaces. As is seen in compact bone, levels of calcificationvary from trabecula to trabecula and within trabeculae.

The connections and orientations of trabeculae are found to have precisepatterns, which are believed to relate to specific mechanicalproperties. The structure of the cancellous bone in the head and in theneck of the femur is usually given as an example of the correlationbetween the orientation of the trabeculae and the linear distribution ofthe principal forces during load bearing (stress trajectoral theory(Bell, 1956)). In general, such correlation between the orientation ofthe trabeculae and the linear distribution of the principal forcesduring load bearing is still under study because while in line with themathematical calculations, the possible effect of muscle traction iscomplex (Koch, 1917; Rybicki et al., 1972). Nevertheless, there is aclose relationship between the number and arrangement of trabeculae andthe strength of cancellous bone (see e.g. Kleerekoper et al., 1985).This is evidenced by the age-induced loss of trabeculae (see e.g.Birkenhäger-Frenkel et al., 1988). Since this loss is rather selective(i.e. transverse trabeculae disappear more frequently than vertical onesin the central zone of the osteoporotic vertebral body; entiretrabeculae totally disappear in elderly women; sharp fall in trabecularnumber is seen in elderly men), it is possible that cancellous bonecontains some bundles of trabeculae whose main function is to resistmechanical forces while others have mainly a metabolic role.

The mechanical behavior of an object, or the response of an object toforces, of an object depends on the structure of the object. If theobject is comprised of a hierarchical structure, the mechanical behaviorof the object varies from order to order. That is, each order or levelof the hierarchy responds to forces according to the structures andrelationships within that order. Overall mechanical behavior of theobject is ultimately determined by the mechanical properties of thedifferent orders. Therefore, the mechanical properties of an object willvary with the hierarchical structure of the object. Bone is an exampleof an object where the mechanical behavior and mechanical properties aredependent upon this kind of hierarchical structure.

Mechanical properties of bone have been and are being investigated atvarious hierarchical levels through invasive (specimen isolation) andnon-invasive testing. Osteonic trabecular lamellae, osteons, trabeculae,and macroscopic compact and cancellous bone samples have been and arethe objects of such studies. Micromechanical results include Ascenzi A.and Bonucci, 1964, 1967; Ascenzi A. and Bonucci, 1968, 1972; Currey,1969; Ascenzi A. et al., 1985, 1997, 1998; Hohling et al., 1990; AscenziA. et al., 1990, 1994; Marotti et al., 1994; Ziv et al., 1996; AscenziM.-G., 1999a, 1999b; Huja et al., 1999; Zysset et al., 1999; AscenziM.-G. et al., 2000. Macromechanical results include Hazama, 1956; Cookand Gordon, 1964; Carter and Hayes, 1976 and 1977; Carter et al., 1976and 1981; Carter and Spengler, 1978; Hayes and Carter, 1979; Burr etal., 1988; Cater and Carter 1989; Jepsen and Davy, 1997.

Viscoelasticity

Material science regards viscoelasticity as a characteristic feature ofpolymer containing materials. Bone, like most biological materials,contains polymers. Viscoelastic properties depend on temperature andmoisture content, which the work will hold constant at physiologicallevel. In viscoelastic materials, some of the elastic energy generatedby application of external forces is dissipated as heat. Such dissipatedenergy may contribute to the force driving the bone remodeling process(Levenston and Carter, 1998).

Studies of the mechanisms that generate bone viscoelasticity, that canshed light on the physico-chemical origin of Wolff's law and theproperties of osteons, have not yet been conducted. However, osteonviscoelastic behavior could differ between longitudinal osteons andalternate osteons at the same degree of calcification (initial andfinal) and within each osteon type between initial and final stages ofcalcification. The only reported mechanical testing of whole osteons,isolated at their natural boundaries, is a preliminary non-systematicmonotonic dynamic torsional loading (Frasca et al., 1981). It indicatesstructure and strain dependence of shear storage modulus in osteons ofunspecified type and degree of calcification, and a linear viscousbehavior up to strain values of 10⁻⁴. However, this model does notexplain the behavior of the osteon structural components.

Even though numerous publications have addressed bone micromechanics inrecent years, many biomechanical issues relating to bone are still notunderstood due to the lack of reliable or predictive models. The lack ofinclusion of such micromechanical properties in current models of bonefunctions and behavior have severely limited their usefulness inpredicting macromechanical properties. These properties include the bonebehavior in response to external forces or identifying the requirementsof bone reconstruction and prosthesis. However, the inclusion of thesefactors requires the development of methods and studies that may providereliable and reproducible results.

There is a need in the art for realistic and meaningful models of bonebehavior. To meet this need, the above-discussed parameters need to beinvestigated and resolved. The studies described herein providesurprising insights into the role of the underlying organizationalarrangement of osteonic lamellar specimens in resisting applied loads.These findings are integrated into a novel and meaningful hierarchicallybased model of the mechanical behavior of macroscopic bone.

Such a model could provide clinicians with a tool to fundamentallyimprove the precision of their interventions.

The present invention describes a method to understand and predict thebehavior of bone. The method includes a model of macroscopic bone whichis constructed in terms of bone's hierarchical structural and mechanicalproperties and their interaction with forces acting on the macroscopicbone, including forces associated with the ordinary functioning of thebody and forces applied clinically. The method can be applied to anybone structures, including human bone and the bones of vertebrates ingeneral. The model applies to normal bone, and to pathological bone,when the pathology either does not alter the structural hierarchy, orwhen the alterations are characterized. The model is also applicable tofossilized bone.

SUMMARY OF THE INVENTION

The present invention contemplates a model of macrostructural propertiesof bone. The model comprises hierarchical structural and hierarchicalmechanical properties of microstructure of the bone and includesinteractions of the bone with internal and external forces. In apreferred embodiment, the bone that is modeled is either compact bone,particularly osteons and their lamellar layers, or cancellous bone. Inan additional preferred embodiment, the mechanical properties used inthe model are selected from the group consisting of tension,compression, shear, bending, torsion, prestress, pinching, and cementline slippage.

The present invention also contemplates methods of predictingdeformation and fractures of bone and for identifying the requirementsof bone reconstruction and prosthesis using the model of the presentinvention.

The present invention further provides a geometric/material model ofhierarchical bone based on the mechanical properties and relativecomponents of osteons. The model is based upon experimental studies ofthe microstructural viscoelastic properties (in terms ofultra-structural properties) of the osteon, the predominantmicrostructural component of adult bone. In one embodiment, the modelcomprises one or more elements selected from the following group: osteonmechanical properties; collagen-bundle orientation; the relative contentof collagen and mucopolysaccharides in osteons; hydroxyapatite content;lacunae, canaliculae, and other porosity fluids within the pores;amounts of osteocytes and osteoblasts, and relative contents of otherproteins.

The invention also provides a method of predicting deformation andfractures of bone using a model based on the viscoelastic properties ofbone or osteons. According to this method, the effect of torsionalloading in terms of microcracking, debonding, breakage, and void growthcan be predicted from a model based on collagen-bundle orientationand/or the relative contents of collagen and mucopolysaccharides inosteons.

In addition, the invention provides a method of identifying therequirements of bone reconstruction and prosthesis using a model of theviscoelastic properties of compact bone. This method could, for example,utilize simultaneous computer simulation, based on the model of theinvention, of the interaction of a patient's bone and the bone used forreconstruction or prosthesis.

The invention further provides a mechanical method for studying osteonspecimens of prescribed shape, dimensions, lamellar type arrangement,lamellar thickness and degree of calcification for viscoelasticbehavior. This method is preferably based on structural and dimensionalanalysis of individual lamellae and controlled dynamic torsional tests.

The invention also provides a model for the properties of bone based onthe relative percentages of collagen and mucopolysaccharides in osteons.Preferably, the model also comprises results from mechanical testing,for example, torsional loading, of osteon specimens of the samespecifications as those subjected to biochemical analysis for collagenand mucopolysaccharide content. In one embodiment, the model takes intoaccount the influence of collagen orientation versus percentage contentsof collagen and mucopolysaccharides to assess the influence of theseconstituents on the viscoelastic response of the osteon specimens. Inanother embodiment, there is a correlation between the distribution ofthe percentages of collagen, mucopolysaccharides with the orientation ofcollagen bundles at both initial and final degree of calcification.

Furthermore, the invention provides a Finite Element Model (FEM). Thismodel reflects the geometry, distribution and orientation of theidentified osteon component elements. In one embodiment, FEMincorporates known structural porosity (canaliculae and lacunae) toproperly reflect the microscopic structure of the single osteon.Preferably, a three-dimensional model containing a large number ofelements is prepared to represent the constituents in sufficient detailso that the results converge. The model can be parametrically exercised,within the limits of known property variation, to allow for biologicalvariations and to study biological effects. This can include variationof the porosity distribution and of the bulk modulus in the porosity toassess the effect of fluid in the structure. Finally, the effects ofthese parametric manipulations on fracture and failure characteristicscan be investigated.

Lamella

The present invention is further based on examinations of the structureof lamellar specimens and their behavior under quasi-static loadingunder tension. These studies provide surprising information on the rolethat the ultrastructural constituents play in the mechanical behavior ofa single lamella.

By describing the limits and extent of the role that the lamellarultrastructure plays in determining lamellar mechanical behavior, theinvention provides an improved understanding of how bone tissue absorbsenergy during quasi-static loading is gained, thereby enabling novelcomputer models of bone behaviour.

The invention also provides important insights into the differences inthe processes of tissue formation for dark and bright lamellae, andpoints to different stimulation of the osteocytes responsible for darkand bright lamellar formation.

In addition, the invention provides a method of identifying therequirements of bone reconstruction and prostheses using a model basedupon the surprising findings regarding the ultrastructural constituentsof lamellae. This method could, for example, utilize simultaneouscomputer simulation, based on the model of the invention, of theinteraction of a patient's bone and the bone used for reconstruction orprosthesis.

The above features and many other advantages of the invention willbecome better understood by reference to the following detaileddescription when taken in conjunction with the accompanying drawings.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1. A schematic representation of the upper third of the tibia;i.c.s. and o.c.s. stand for inner and outer circumferential systems,respectively. Both compact and cancellous bone are represented (fromBonucci, 2000, Basic composition and structure of bone. In: MechanicalTesting of Bone (An Y. and Draughn R. eds.) pp. 3-21, CRC Press, BocaRaton, Fla.).

FIGS. 2 (a) and (b). (a) Diagram of a diaphysis sector of cortical longbone. The osteons or haversian system (HA) are located between the outerOL and inner IL circumferential lamellae. The osteonic lamellae aredisposed cylindrically around the haversian canal (HC) (from Bouligandet al., (1985) Spatial organization of collagen fibrils in skeletaltissues: analogies with liquid crystals. In: Bairati A, Garrone R (eds.)Biology of invertebrate and lower vertebrate collagens. PlenumPublishing Corporation). (b) Cross-sectioned osteons as seen (A) under alight microscope; (B) in a microradiograph; and (C) under the polarizingmicroscope (from Bonucci, 2000).

FIGS. 3 (a) and (b). (a) Section of the body of a lumbar vertebrashowing vertical and horizontal trabeculae. The upper and lower surfacescorrespond to articular cartilage (from Bonucci, 2000). (b) Section ofhalf of tibia's upper third. The cancellous bone of the metaphysisconsists of comparatively think vertical trabeculae connected by thintrabeculae (from Bonucci, 2000).

FIG. 4( a)-(c). (a) Cross section of an isolated longitudinal osteon,magnified 270 times. (b) Cross section of an isolated alternate osteon,magnified 270 times. (c) An isolated osteonic sample with lugs,magnified 20 times. Lugs are used to grab the sample during mechanicaltesting. Dimensions: inner diameter 52 μm, outer diameter 225 μm, length500 μm.

FIG. 5. Cross-section diagram of an osteon sample illustrating thearrangement of canaliculae and lacunae relative to lamellae.

FIG. 6 (a)-(c). (a) Types of pure forces; (b) Definition of stress on anarea on which the force is constant; (c) Definition of unidirectionalstrain for D much smaller than L (from Evans F. G., MechanicalProperties of Bone, Thomas, Springfield, 1973).

FIGS. 7( a) and (b). (a) Tensile and compressive stress distributionduring torsion in a material, such as macroscopic bone, which is weakerin tension than in shear; (b) Shearing stress on the cross section of aspecimen subjected to torsion. The arrows' length indicates themagnitude of the shearing stress, which progressively increases from thecenter to the periphery of the specimen (from Evans, 1973).

FIGS. 8(A) 8(B) 8(C) and 8(D). 8A Bending of femur due to gravity. Cindicates the area under compression and T indicates the area undertension. FIGS. 8B, 8C, and 8D display the distribution of transverse andlongitudinal lamellae in the sections prepared from the upper, middleand lower shaft, respectively. The posterior, anterior, medial andlateral regions correspond to the top, bottom, left and right regions,respectively, on the page. The distance between the centers of twoadjacent square symbols measures 1.86 mm. The size of the square symbolis proportional to the ratio of the bright area in circularly polarizedlight to bright area in a dark field illumination. The regions withdominant transverse lamellae correspond to the regions withconcentration of larger squares in the upper medial, middlemedial-posterior and lower posterior shaft, which correspond to theareas of compression in FIG. 8A. The regions with dominant longitudinallamellae correspond to the regions with concentration of smaller squaresin the upper lateral, middle lateral-anterior, and lower anterior shaft,which correspond to the areas of tension in FIG. 8A.

FIG. 9 (a)-(c). (a) The osteonic lamellar model is a laminate, whichconsists of fiber-reinforced unidirectional laminae. (b) Theinterstitial lamellar model is a portion of the osteonic lamellar model.The figure shows three thin laminae (lamellae) and a thick lamina(portion of cement line) (from Crolet, J. M., Aoubiza, B., and Meunier,A., Compact bone: numerical simulation of mechanical characteristics. J.Biomechanics, (26):677-687, 1993). (c) On a small laminar element ofconstant thickness, the principal material axes are labeled 1, 2, and 3.Direction 1 is parallel and direction 2 is perpendicular to the fibers.Direction 3 is the radial direction perpendicular to the page.Circumferential and axial directions are labeled Θ and z. The anglebetween the circumferential direction and direction 1 is called γ (fromAscenzi, M.-G. (1999) A first estimation of prestress in so-calledcircularly fibered osteonic lamellae, J. Biomechanics, (32): 935-942).

FIG. 10. Shows a device for subjecting bone to torsional cyclicalloading (from Ascenzi, A, Baschieri, P. Benvenuti, A. (1994) Thetorsional properties of single selected osteons. J. Biomechanics, 27(7):875-884).

FIG. 11. Is a schematic diagram of a device for subjecting bone totorsional cyclical loading, where (1) is a rotational axis with jaws;(2) and (3) are hard metal wedges of a pendulum loading system; (4) is awheel around which a tungsten thread loaded with weights is attached;(5) is the axis of the pendulum; and (6) is a mirror (from Ascenzi,1994).

FIG. 12A is an idealized bilinear hysteresis model of curve prior tocycling and a first cycling ioop where pinching is present. FIG. 12B isan idealized bilinear hysteresis model of curve prior to cycling and afirst cycling loop where pinching is not present.

FIG. 13. Shows that around each osteon sample, a trapezoid was cut witha blade under a stereo microscope (from Ascenzi, M.-G, Ascenzi, A.,Burghammer, M., Panzavolta, S., Benvenuti, A. and Bigi, A. (2003)Structural differences between “dark” and “bright” isolated humanosteonic lamellae. J. Structural Biology, 141, 22-33).

FIG. 14. Shows that after isolation, each lamellar sample was carefullystraightened to a ribbon-like shape (from Ascenzi, A, Benvenuti, A,Bonucci, E. (1982) The tensile Properties of single osteonic lamellae:technical problems and preliminary results. J. Biomechanics, 15(1):29-37).

FIG. 15. Represents a larger view of the lamella described in FIG. 14(from Ascenzi, 1982).

FIG. 16. Shows a lamella after tensional testing (from Ascenzi, 1982).

FIG. 17. Shows collagen bundles of bright lamella under polarizedmicroscope.

FIG. 18. Shows collagen bundles of bright lamella under polarizedmicroscope.

FIG. 19. Shows collagen bundles of extinct lamella under polarizedmicroscope. Bundles are parallel to the osteon axis when embedded inbone.

FIG. 20 (a)-(h). Shows isolated and flattened bright lamella under theconfocal microscope. From border to center, the collagen bundles go fromoblique to vertical (from Ascenzi, M.-G, Ascenzi, A., Burghammer, M.,Panzavolta, S., Benvenuti, A. and Bigi, A. (2003) Structural differencesbetween “dark” and “bright” isolated human osteonic lamellae. J.Structural Biology, 141, 22-33).

FIG. 21 (a)-(g). Shows isolated and flattened extinct lamella under theconfocal microscope. From one border to the other, the collagen bundlesare parallel to the osteonal axis (from Ascenzi, 2003).

FIGS. 22A, 22B, and 22C. (A) Cross section of isolated longitudinalosteon (outer diameter: 225 μm). (B) Cross section of isolated alternateosteon (outer diameter: 225 μm). (C) Isolated osteon sample with lugsfor torsional loading (inner diameter: 52 μm; outer diameter: 225 μm;length: 500 μm).

FIG. 23. Schematic drawing of torsional loading device used in theExamples. 1=Rotational axis with its jaws 2; 3=hard metal wedges of thependulum loading system; 4=the wheel around which the thread, lodgedwith weights, is attached; 5=the axis of the pendulum; 6=the mirror thatreflects the laser beam onto the graduated scale to detectangle-of-twist variations (from Ascenzi, 1994).

FIG. 24. Diagram showing the trapezoid cut from a thin transversefemoral section around a chosen alternate osteon (from Ascenzi, 2003).

FIG. 25. Isolated osteon sample (under transmitted light) for theproposed biochemical analysis (length 500 μm).

FIG. 26. Diagram of a segment of osteon in cross-linking illustratingthe arrangement of canaliculae and lacunae relatively to lamellae (basedon FIG. 4-11 of Leeson et al. (1985)).

FIGS. 27A and 27B. (A) Material model consisting of fiber-reinforcedunidirectional laminae. The first few external laminae are partiallypulled out to show arrangement. (B) On a small laminar element ofconstant thickness, the principal material axes are labeled 1, 2, and 3.Direction 1 is parallel, and direction 2 perpendicular, to the fibers.Direction 3 is the radial direction, perpendicular to the plane of thediagram. Circumferential and axial directions are labeled θ and z. Theangle between the circumferential direction and direction 1 is denoted y(from Ascenzi, M.-G. (1999) A first estimation of prestress in so-calledcircularly fibered osteonic lamellae, J. Biomechanics, (32): 935-942).

FIGS. 28A and 28B. (A) Cross section of two isolated osteons:longitudinal (above) and alternate (below), ×110. (B) Osteon specimen(inner diameter: 40 μm; outer diameter: 210 μm; length: 500 μm) with itslugs, ×30.

FIG. 29. Isolated and flattened bright lamellar specimen rotated at 45°to the polarizing plane. Collagen bundles run parallel to its length.Lamellar width is approximately 70 microns.

FIG. 30. Isolated and flattened bright lamellar specimen rotated at 0°to the polarizing plane. Collagen bundles run parallel to its length.Lamellar width is approximately 70 microns.

FIG. 31. Confocal microscopy detail of isolated and flattened darklamella. Opposed arrows show the orientation of a collagen bundlearrangement perpendicular to the lamella edge. Cut radial collagenbundles appear as dots within the circle.

FIG. 32. Detail of isolated and flattened bright lamellar sample asviewed by confocal microscopy. Arrows indicate oblique bundles extendingacross the lamella thickness.

FIG. 33 A Diagram of relative inclination of dark lamellac with respectto incident beam. FIG. 33B Diagram of relative inclination of brightlamellae with respect to incident beam. Dots indicate chosen locationsfor scanning.

FIG. 34. SAXS images of dark lamella. The images are unchanged acrossthe scanned area.

FIG. 35. Enlargement of one SAXS image from FIG. 9. Clear arching andmaximum intensity orientations show single preferential collagen bundledirection perpendicular to bright lamellar width.

FIG. 36. WAXS image of the scanned location of the FIG. 10 SAXS image.Clear preferential orientation of the 002 reflection parallel to thedark lamella width shows single preferential collagen bundle directionperpendicular to bright lamellar width.

FIG. 37. SAXS image of dark lamella from a scanned area. The imageschange across the scanned locations.

FIG. 38. WAXS image of a bright lamella scanned location, which showslack of preferential orientation of the 002 reflection.

FIG. 39 Diagram showing the trapezoid cut from a thin transverse femoralsection of bone that contains a chose alternate osteon (from Ascenzi,2003).

FIG. 40 Dark lamellar specimen oriented at 0° (top) and 45° (bottom)with respect to the plane of polarization.

FIG. 41 From top to bottom: the orientation of the bright lamellarspecimen changes in discrete steps of 0°, 39°, 45°, 80°, 84°, 90°relative to the plane of polarization.

FIG. 42 Confocal microscopy detail of isolated and flattened darklamella. Opposed arrows show the orientation of a collagen bundlearrangement ⊥ to the lamella edges. Cut radial collagen bundles appearas dots within the circle.

FIG. 43 Detail of isolated and flattened bright lamellar sample asviewed by confocal microscopy. Arrows indicate oblique collagen bundlesextending across the lamella thickness.

FIG. 44 Diagram of relative inclination of dark (a) and bright (b)lamellae with respect to the incident beam. Dots indicate chosenlocations of scanning.

FIG. 45 SAXS images of dark lamella. The images are unchanged across thescanned area.

FIG. 46 Enlargement of one SAXS image from FIG. 45. Clear arching andmaximum intensity orientations show single preferential collagen bundledirection perpendicular to bright lamellar width.

FIG. 47 WAXS image of the scanned location of the FIG. 48 SAXS image.Clear preferential orientation of the 002 reflection parallel to thedark lamellar width shows single preferential collagen bundle directionperpendicular to bright lamellar width.

FIG. 48 SAXS images of dark lamella from a scanned area. The imageschange across the scanned locations.

FIG. 49 Photograph of lamella specimen attached to SEM grid sections asprepared for the tensile test.

FIG. 50 On a small and thin laminar element, the principal material axesare labeled 1, 2, and 3. Direction 1 is parallel, and direction 2perpendicular, to the fibers. Direction 3 is the radial direction,perpendicular to the plane of the diagram. circumferential and axialdirections are labeled θ and z. The angle between the circumferentialdirection and direction 1 is denoted by γ (from Ascenzi, M.-G. (1999) Afirst estimation of prestress in so-called circularly fibered osteoniclamellac, J. Biomechanics, (32): 935-942).

FIG. 51 Isolated flattened lamellar specimen. The breadth of this figurecorresponds to the “length” of the lamellae.

FIGS. 52A and 52B. (A) SAXS of dark lamella; (B) WAXS of dark lamella.

FIG. 53 SAXS of bright lamella.

FIG. 54 SAXS of bright lamella.

FIG. 55 SAXS of bright lamella.

FIG. 56 Collagen and hydroxyapatite dominant directions of dark andbright lamella.

FIGS. 57A and 57B. X-ray diffraction results of (A) dark lamella and (B)bright lamella.

FIG. 58. Twisted plywood model of osteon.

FIG. 59 Dark and bright lamellar distribution correlates to cyclicforces and geometry.

FIG. 60( a)-(d). Examples of biological variations within transverse(a-c) and longitudinal (d) osteon sections viewed by transmissionmicroscopy. (a) Extinct osteon #551 (×281); (b) alternate osteon #311(×281); (c) bright # 381 (×281); and (d) alternate osteon #361 (×637).

FIG. 61( a)-(b). (a) Angle ranges of collagen-hydroxyapatite orientationfor extinct (−45°, 45°) and bright (45°, 135°) with respect to theosteon axis. The circled values 90° and 0° are the dominantorientations. (b) Collagen-hydroxyapatite orientation angle within twoadjacent extinct and bright lamellae of alternate osteon within lamellarthickness of 6.25 and 10 μm respectively. The horizontal segments show adominant orientation at 0° for the extinct lamella and minor orientationat 90° for the bright lamella.

FIG. 62( a)-(b). (a) The centers of the ellipsoids which model thelacunae are first distributed randomly along n solid rays (in thismodel, n=16) and then moved circumferentially randomly within the regionbracketed by the adjacent dashed rays. (b) The generated ellipsoids thatmodel the lacunae within the two cylindrical surfaces that lie next tothe Haversian canal and cement line.

FIG. 63( a)-(c). Details of the lacunar-canalicular network in the modelof bright osteon #381: (a) transverse view (×281); (b) 3-dimensionalview; (c) a three-dimensional detail of the sector model with a planeindicating the section represented in (b). The (two) lacunae and theircanaliculae that intersect the plane appear also in (b) while the restof the structure is randomly generated.

FIG. 64( a)-(c). Alternate osteon specimen #361 viewed longitudinally(×637): (a) photograph; (b) model of extinct and bright lamellae only;and (b) model of extinct and bright lamellae with lacunae. (c) Alamella-lacunar model in providing a representation of the three lacunaeinside the square.

FIG. 65( a)-(b). Example of lamellar isolation simulation. (a) Lamellarmodel obtained from (b) alternate osteon model by intersection.

FIG. 66( a)-(e). Ellipses obtained by intersecting ellipsoids (whichrepresent lacunae not to scale) with transverse planes at a constantvertical distance from the centers of the ellipsoids. The area of theellipse increases with the angle between the major axis of the ellipsoidand the vertical direction. The angle equals (a) 0°; (b) 22.5°; (c) 45°;(d) 67.5°; and (e) 90°.

FIG. 67( a)-(b). Confocal microscopy images of (a) an extinct lamellaspecimen, and (b) a bright lamella specimen with characteristic bundlessomewhat paralle to specimen length.

FIG. 68( a)-(f). Confocal microscopy images of bright lamellae, showingcollagen orientation.

FIG. 69. Confocal microscopy images of extinct lamellae, showingcollagen orientation.

DETAILED DESCRIPTION OF THE INVENTION

The present invention describes a method for modeling the anisotropic(direction-dependent) and non-homogeneous macrostructural properties ofcompact bone in terms of the microstructure. The model is based on thehierarchical structural and mechanical properties and bone interactionswith internal and external forces. An example of such forces includes,but is not limited to, the ordinary functioning of the body. The modelincludes properties of the microstructure, in particular distributionsof transverse lamellae of trabeculae and of alternate osteons, pinchingof osteons, and slippage of osteons at the cement line.

Morphological and mechanical studies of bone show that at allhierarchical levels bone is anisotropic (the local mechanical propertiesare direction dependent), and non-homogeneous (the structure is not thesame at different points). Nevertheless to simplify bone modeling, bonestructure often is assumed to be homogeneous, isotropic (not directiondependent), transversely isotropic (one plane of symmetry), ororthotropic (three planes of symmetry). The simplifications of isotropy,orthotropy, and transverse isotropy give rise to unrealistic modelsbecause these simplifications assume that symmetries that do not exist.For instance, in such models stresses may be over- or under-estimated.When such models are applied to practical applications, for example boneimplants, poor estimates of stress may give rise to screw loosening inimplants. The simplification of homogeneity gives rise to unrealisticmodels because it disregards the hierarchy of the bone structure. Theexisting hierarchical models are based on homogeneity theory, finiteelement analysis, and classic and Cosserat elasticity theories (see e.g.Katz, and Meunier, 1987; Crolet et al., 1993; Pidaparti and Burr, 1992).These models do not include important properties of the microstructure,which are included in the present invention and are described below.

The present model provides for modeling each level of the hierarchicalstructure of bone in terms of the structural and mechanical propertiesof that level. The model further provides for determining therelationships among the various levels.

In compact bone:

(1) Collagen bundles, hydroxyapatite crystallites, andmucopolysaccharides are organized in two lamellar types, bright, whichare prestressed, and dark extinct lamellae. Lamellae show porosity.

(2) Lamellae are organized in osteons. Osteons show pinching undertension-compression cyclic loading.

(3) Osteons samples are organized in osteon sample groups on bonesections. Osteon groups show cement line slippage during torsion.

(4) Osteon groups are organized to complete a bone section. The collagenbundle direction distributions are used to complete this organization.

(5) Transverse sections are organized to complete the macroscopic bone.

In cancellous bone:

(1) Collagen bundles, hydroxyapatite crystallites, andmucopolysaccharides are arranged in bright and dark extinct lamellae.Bright lamellae are prestressed. Lamellae show porosity.

(2) Lamellae fill in trabeculae. Trabeculae show porosity.

(3) Trabeculae are grouped in trabecular sample groups on bone sections.

(4) Trabecular groups are organized to complete a bone section. Theprevalent collagen bundle directions are used to complete thisorganization.

(5) Sections are organized to complete the macroscopic cancellous bone.

To produce the model, each microstructural level of the hierarchicalbone structure commences with the micro-structural components andproceeds through the macro-structure. Each element of the assembly iscorrelated with mechanical properties either determined from literaturesources or that are newly estimated. Homogenization methods are used toassemble the structure at one level with the structure at the nextlevel, and so on, to build a hierarchical model. The finite elementmethod allows for the computation of strains and stresses throughout themodel.

The invention results in a model of macroscopic compact and cancellousbone that respects the hierarchical, structural, and mechanicalproperties starting from the microstructural components. The model maybe applied to all bones to result in a model for each bone of theskeleton.

The present invention further defines methods of predicting deformationand fractures of bone and identifying the requirements of bonereconstruction and prosthesis using the model. From the specific forcesthat act on bone during function, the model allows for the assessment ofbone deformation, strain, stresses, and fractures. Additionally, fromthe fractures and stress distribution, the model allows for thecomputation of strain deformation and forces that cause the observedfracture and stress distributions. The model also contemplates forcesthat do or do not cause viscous effects. The model contemplates forcesthat cause either elastic or plastic bone deformations as characterizedby fractures.

The model includes torsional cyclic loading functions of tworepresentative osteon types in terms of degrading properties such asstiffness and pinching, and increasing energy absorption. Thesemechanical property changes are correlated to the idealized ormathematical behavior of ultrastructural components, which includesyielding, buckling, and fracturing properties. The resulting algorithmsand behaviors comprise an osteon model, which simulates fracturepropagation in osteons under cyclic torsional loading in terms ofmicrocracking, debonding, void growth, and fiber breakage. Verificationof the model is demonstrated by checking that the model produces thefractures observed in osteon samples that are separately submitted totension, compression, and shear.

The model also includes simulation of microstructural fracturepropagation in bone. Because of the dependence of the macrostructure'smechanical properties on microstructure, the model will provide animproved understanding of properties of long bone, such as fracturepropagation, including a better understanding of how human bonemacrostructure responds to forces acting on it.

The model will have application in many areas, including withoutlimitation:

-   -   the mechanics of natural composites and the manufacture of new        composites, since bone is a natural composite material;    -   the identification of the fundamental requirements of bone        reconstruction and prostheses (which will increase design        effectiveness and reduce testing and related cost); and    -   the microstructure of vertebrates whose microstructure is        similar to human's

Definitions

The present invention spans through both elastic and plastic ranges. Asused herein, the term “elastic range” refers to the stress and strainvalues for which the material structure does not break and returns toits original shape when the force is removed. As used herein, the term“plastic range” refers to the stress and strain values for which thematerial structure does break and therefore does not return to itsoriginal shape when the force is removed. When an increasing force(starting from zero) is applied to a material, the material undergoesfirst elastic and then plastic deformation. Any bone type can undergoelastic deformation only or both elastic and plastic deformationdepending on the force magnitude. Elastic and plastic deformationsprovide a starting point to predict strain and stress distributions andfractures of bone. The model also may be used to compute the stressdistribution from the strain distribution and strain distribution fromelastic and plastic deformations. It further identifies the requirementsof bone reconstruction and prostheses.

As used herein, the term “boundary conditions” refers to the relativemovements of the boundaries of the various hierarchical structures underloading. In a specific embodiment, the behavior of the cement line underloading is the boundary conditions for the osteon and the interstitialbone between which the cement line lies.

The term “pinching” refers to a sharp change of stiffness of bone. Asused herein, the change in stiffness can be either from increasingstiffness to decreasing stiffness or from decreasing stiffness toincreasing stiffness. In a preferred embodiment, the change presentsitself on each half-cycle.

The term “material analog” refers to a model or reproduction producedfrom material, as distinguished from a mathematical or computer model.

The term “distraction device” refers to an apparatus that generates boneby stimulating growth of existing bone by application of forces to suchexisting bone.

As used herein, the term “strain distribution” refers to a measure ofthe degree of elongation at any point on a sample. In a preferredembodiment, the sample is bone.

The term “stress” refers to the force per unit area.

The term “stress distribution” refers to strain distribution and on themechanical property distribution throughout the body.

The term “corrected break area” refers to the actual bone area, exceptfor the lacunae and canaliculi, subjected to stress in the vicinity of abreak.

The terms “viscous effect” and “viscoelastic” refer to a system thatexhibits behavior that combines liquid-like and solid-likecharacteristics. In particular, this term herein refers to theviscoelastic properties of osteons. According to the invention, theseviscoelastic properties include one or more of the following parameters:osteon mechanical properties (e.g., elastic and viscous constants forosteon specimens, torque versus angle-of-twist behavior, etc.), degreeof calcification of osteons, the relative contents of collagen andmucopolysaccharides in osteons, collagen-bundle orientation in osteons,osteon hydroxyapatite content, amount of lacunae and canaliculaeporosity fluids within the pores, osteon content of osteocytes andosteoblasts, as well as the contents of other proteins present inosteons. Preferably, these parameters are measured in both longitudinaland alternate osteons.

A Finite Element Model, or “FEM”, is a well-known method for modeling.According to this model, the gross shape of the object, e.g., a bone,that one wishes to model is filled with homogenous elements (e.g., 3Dbrick-shaped) in a finite number. The mechanical analysis is conductedon the individual elements, so that elastic or viscoelastic propertiesare declared for each element. The loads and boundary conditions areapplied to the gross shape through the distribution of loads on theelement. At this point, one can compute the strain and stress for eachelement. A sufficient number of elements must be applied so that thestrain and stress distributions are compatible along the boundariesbetween any two adjacent elements. The principles of FEM has beendescribed, e.g., in Dwoyer et al., (1988), which is hereby incorporatedin its entirety.

A “reference osteon” is a set of osteon data and/or parametersrepresentative of an osteon to be used in the model of the invention.

Factors

Mechanical Properties

Various mechanical properties are included in the model of the presentinvention. The properties will be correlated with each hierarchicallevel of the bone to produce the model. A non-limiting list of suchproperties is disclosed and described herein.

The mechanical properties of bone are quantified by parameters orcoefficients that describe the response of bone to tension, compression,shear, bending and torsion. Tension, compression, and shear are termed“pure forces” because each of them is recognized by the effect (thedeformation) produced in the body to which it is applied (FIG. 6). Atension (tensile) force tends to lengthen the body to which it isapplied, while a compression (compressive) force has a tendency toshorten the body. A shear force tends to make one part of a body slidein a direction opposite to that of an adjacent part. Bending and torsion(FIG. 7) are a combination of tension, compression and shear.

The effect of the application of one of the above-mentioned forces to abody at a natural state is described in terms of strain and stress.Strain is the measure of dimensional changes in a body and is computedby means of the deformation (Antman, 1995). Since in general the valueof strain changes from point to point throughout the body, more properlyone refers to strain as the strain distribution throughout the body,which provides the value of strain at each point of the body. Thetendency of a body to be deformed by the application of a force isresisted by the internal force among the molecules composing the body.Such resistance is measured by the stress, which is a force per unitarea. Similar to strain, in general the value of stress changes frompoint to point throughout the body. More properly one refers to stressas the stress distribution throughout the body, which provides the valueof stress at each point of the body. The stress distribution depends onthe strain distribution and on the mechanical property distributionthroughout the body. What all elastic structures have in common is thatthe stress distribution is a linear function of strain within theelastic range (Hooke's Law; see e.g. Jones). Beyond the elastic rangethe relationship between stress and strain distributions depends on theparticular structures. For instance, the mechanical testing of aspecimen provides a stress-strain diagram, which allows the study of therelationship between stress and strain.

Studies indicate that the mechanical behavior of longitudinal andalternate osteon samples at equal degree of calcification, as assessedby the method of Amprino and Engström (1952), differs because of theirstructural difference. The comparison of experimental stress-straindiagrams for longitudinal and alternate osteons shows that undermonotonic tension and torsion, longitudinal osteon samples resiststresses better than alternate osteons; while under compression,shearing and bending, alternate osteon samples resist stresses betterthan longitudinal osteons. Under cyclic tension-compression loading,longitudinal osteon samples show a larger energy loss and lesserpinching degradation per cycle than alternate osteons; longitudinalosteon samples show a greater strain increase during compression thantension. The opposite is true for alternate osteons.

The macroscopic mechanical properties have been found to depend on andto be explained by the microstructure. In particular, they have beenfound to depend on the numerical presence of osteons, the size andpercentage volume of osteons, and collagen fiber orientation (Currey,1959; Evans and Vincentelli, 1969). As early as 1873, Rauber consideredthe correspondence between bone micro- and macro-structure. Hehypothesized that the structure of osteons and interstitial bone in thelong bone shaft relates both to their distribution in the shaft undernormal conditions, and also under pathological conditions that do notalter the bone's hierarchical configuration. This hypothesis was laterconfirmed (Portigliatti-Barbos, 1983, 1984, and 1987; Boyde et al.,1984; Ascenzi A. et al., 1987a and 1987b; Ascenzi A., 1988; Carando etal., 1989 and 1991). Specifically, the distribution of dark lamellae(whose bundles have a transverse and oblique course) and of brightlamellae (whose bundles have a longitudinal course) in osteonic andinterstitial bone follows a characteristic non-random pattern. Studiesindicate that this distribution is consistent with the distribution ofbending forces usually operative on this bone (Ascenzi M.-G., 1999a).For example, the femoral dominant distribution of dark and brightlamellae displays a clockwise rotation of approximately 90° insequential sections from upper, middle, and lower third of the shaft(Portigliatti-Barbos, 1983, 1984). In fact, because of the femoraloverall shape which includes two curvatures (an anterior-posteriorcurvature and a lateral-medial curvature), gravity on the body resultsin the bending of the femur. Because bending always includes an area oftension and an area in compression, the femur presents an area intension and an area in compression (FIG. 8 a). It turns out that thefemoral dominant distribution of dark (bright, respectively) lamellaecoincides with the area in tension (compression, respectively) (FIG. 8b). Recent work also has found that the above-mentionedtransverse/longitudinal lamellar distribution is consistent with thedistribution of alternate osteons (Hert et al., 1994). Neither thetransverse/longitudinal lamellar distribution nor the alternate osteondistribution have been included in prior models of bone structure.Transverse/longitudinal lamellar distribution and the alternate osteondistribution are included in exemplary models of the present invention.

Additional Factors

The following provides a non-limiting list of factors that may beincluded in the models of the present invention, and which are used inexemplary embodiments.

(1) Fracture of macroscopic bone. The invention incorporates fracturedynamics into the bone model and modeling methods, including mechanismsby which a fracture starts and spreads. Unlike other models of bone,fracture propagation is modeled in terms of ultrastructural components.The literature indicates that the fracture mechanism of bone depends onbone structural and composition properties such as collagen architectureand collagen content (e.g., Jepsen et al., 1999). In 1969, Evans andVincentelli showed significant differences among osteons of variousbones (fibula, tibia, and femur) in the “corrected break area”, which isthe actual bone area, except for the lacunae and canaliculi, subjectedto stress in the vicinity of a break. Characteristic differences werefound between the means of the corrected break area for groups oflongitudinal and transverse (i.e., consisting of transverse lamellae)osteons and of osteon fragments of the femoral and tibia sections andfor groups of the transverse osteons and fragments of the tibia andfibula sections. The percentage of the “corrected break area” oftransverse osteons and their fragments in the tibia and fibula sectionswas also statistically different. Another study (Vincentelli and Evans,1971) established a relationship among macro-mechanical properties,collagen bundles, and calcification in the shaft of long bones.Furthermore, fracture lines appear to follow the cement lines betweenosteons and lamellar boundaries within osteons (Simkin and Robin, 1974)where the bone is weaker. According to the invention, inclusion of thedifferences between the means of the corrected break area for groups ofosteons would increase the predictability of the present model comparedto prior art models.

(2) The prestress distribution in bone. The models and methods of theinvention incorporate computations of the stress distribution in longbone so as to include pre-stress (Currey 1964, Ascenzi A. and Benvenuti,1980). Stress distribution in long bone depends on structural andcomposition properties such as collagen architecture and collagencontent. Bone areas where collagen bundles are transverse and oblique tothe long bone axis are prestressed. Such prestress, estimated on theorder of 0.1 GPa, is too large to be disregarded. It locally impacts thestress produced by forces acting on bone (see Ascenzi M.-G., 1999a).Newly estimated prestress variables are included in the presentinvention (Ascenzi M.-G., 1998a and 1999b). The newly estimatedprestress was evaluated through the structural and mechanical modelingof isolated lamellar samples, and has been shown to be a realisticapproach. See e.g., A. Meunier, 1999. Inclusion of this prestress intothe model of the present invention allows one to more accurately modelbone in terms of computation of stresses.

(3) The phenomenon of “pinching”. The invention for the first timeincorporates pinching into bone models and modeling methods. Pinching isthe mechanism of yielding and buckling of collagen bundles under loadingbeyond the elastic phase. It is an important step in the formation andpropagation of fractures. The understanding of pinching requiresdetailed analysis of osteon mechanical behavior. In fact, while thestress-strain curves for monotonic loading under tension, compression,and torsion show trends no different from those recorded frommacroscopic bone samples, the tension-compression hysteretic loopsshowed a new behavior for osteon samples not observed in macroscopicsamples (Ascenzi et al., 1985 and 1997). The new behavior observed isthat tension-compression hysteretic loops of osteons demonstrateS-shaped half-cycles. This phenomenon has been observed and studied onlyrelative to earthquake-resisting structures. In such context thebehavior is usually called “pinching” (see, e.g. Narayanan and Roberts,1991). Pinched hysteretic loops are typical of structures thatincorporate a matrix that cracks and reinforcements that yield or whosemembers buckle when subjected to compressive loads. In osteons, theshape and dimensions of hydroxyapatite crystallites and the relationshipof these parameters to the organic components of the matrix are onlypartially known. Not all collagen bundles are completely calcified.Those that are not calcified take up crystallites only on 400 Å bands(Ascenzi A. et al., 1965). Hence such bundles may be comprised ofrelatively more stiff 400 Å bands separated by relatively more flexiblenon-calcified collagen segments.

Pinching in osteons is hypothesized to be mainly localized at thepartially calcified bundles. Therefore, in osteons, either bundles yieldin tension and buckle in compression while crystallites fracture anddetach from collagen, or crystallites fracture and detach from collagenin tension while collagen yields in compression. Thus, cyclictension-compression loading shows pinching. Since cyclic torsionalloading involves tension and compression, cyclic torsional loading isexpected to show pinching. Nevertheless, it may be that the disruptioncreated by torsional loading is too disordered, in comparison to thatdue to tension-compression, to allow closing of lesions and resolutionof members as controlled as under tension-compression. In any event, ifcyclic torsional loading of osteons shows pinching, pinching is includedin the invention applied to macroscopic compact bone torsional loading.

(4) Macrostructure and mechanical loading studies of whole bone ormacrosamples. The invention takes account the influence of bonemicrostructure in evaluating mechanical loading of whole bone and ofbone macrosamples. In the literature, for example the torsional loadingin bone has been analyzed using finite element analyses (see e.g.Hazama, 1956; Pfafrod et al., 1972 and 1975; Knets et al., 1973; Millerand Piotrowski, 1974; Evans, 1978; Martens et al., 1980; Moreland,1980). However, models presently do not completely reflect the changingproperties of bone at the microstructural level. Similarly, cancellousbone has been described as continuous and isotropic, which does notreflect the high porosity and the changing details (such as collagenbundles direction and lamellar structure) at the microstructural level.The elastic and plastic moduli change locally in relation to themicrostructural properties.

Such studies ignore most of the mechanical properties of themicrostructure (because macroscopic samples do not always have the samemechanical properties as the microstructure that comprises them) andtherefore do not provide a realistic understanding of bone mechanics.For instance, pinching is present in longitudinal and alternate osteonsbut not in macroscopic compact bone samples during tension-compressioncyclic loading; also the torsional stiffness varies from osteon samplesto osteon groups and relative to that of larger compact bone samples(Lakes, 1995). Lakes shows that the torsional shear moduli of osteonsare much larger than shear moduli obtained for macroscopic samples. Thatis, slender specimens are stiffer than thick ones; the lower stiffnessin thick specimens is attributed to slippage of osteons at the cementlines during torsion of macrosamples. Such slippage is described well byCosserat elasticity theory since it allows a moment per unit area inaddition to the usual force per unit area of classic elasticity theory.The inclusion of this factor into the model impacts, for example, thesimulation of fracture propagation. The fracture propagation model isable to simulate the slippage of osteons at the cement lines duringtorsion and therefore the experimentally obtained results regardingfractures spreading along the cement line.

Local Properties and Bone Modeling

The knowledge of the mechanical properties and of the strain and stressdistributions of compact and cancellous bone under specific loading isnecessary in all contexts where the local behavior of bone is inquestion. For example, stability is the crucial characteristic of anosteotomy fixation device. When a tibia requires an osteotomy, thedevice that holds in place the two bone edges created by the cut(osteotomy) can only allow for micro-movements of one edge with respectto the other during function, such as walking, to be successful. Thestability of device depends on its shape, and material, and on thenumber, position and inclination of the pins that secure the device tothe tibia. The best position and inclination of pins for the stabilityof the device depends on the spot chosen, that is on the local propertyof the tibia. The anisotropy and non-homogeneity of the tibia make adifference with respect to the screw loosening while walking. In fact,the screw may or may not get loose if the chosen spot is more or lessresistant to the force that it takes for the pin to get into place, ifone inclination is chosen instead of another one, if the spot isprestressed in one direction or another. The question of osteotomyfixation stability cannot be fully studied with a computer bone modelthat does not take into account the hierarchical structure of bone thatrenders bone anisotropic and non-homogeneous. Because if the modelassumes less than that, the local information is lost and the bone showsthe same properties where it should not.

Another example refers to cemented implants. The local bone conditionsaffect the bone-implant interface. The loosening rates in cementedimplants, especially in younger, active persons, is partially due to thelocal bone mechanical properties. This problem has led manyinvestigators to pursue methods of cementless fixation. In the meantimea great deal of attention is being focused on the bone-implant interfaceand the factors affecting its strength. A thorough solution to theproblems involves the knowledge of the local bone mechanics.

Simulation of Fracture Propagation

The fracture propagation model of either compact or cancellous boneunder specific loading follows the same steps as the fracturepropagation model of single osteon samples under torsion (See, Example2, Steps 1-19). The computer program may be based on any suitablesimulation program, for example, a Monte Carlo simulation. The fracturepropagation steps are applied to the finite element mesh for the compactor cancellous bone in question, instead of to the finite element mesh ofsingle osteon samples.

The purpose of the fracture model is to show that cumulativemicro-cracking, de-bonding, void growth and fiber breakage associatedwith repeated loading of osteons causes a progressive loss of stiffnessand pinching, and increase of energy absorption.

The fracture model reflects the following hystological/physiologicalobservations. Fluids occupy vascular canals, canaliculae and lacunae,which are interconnected. The flow of liquids under stress can absorblarge amounts of energy, increasing the toughness of bone. Large strainsmay be accommodated by the organic phase (e.g. collagen,mucopolysaccarides). When a strain is sufficient to cause cracking, theorganic phase may also contribute to the dissipation of energy at thefront of a propagating crack. Crack propagation also appears to bearrested in the presence of canaliculae and lacunae. In fact, when thecrack gets to a hollow space, it just stops because at the hollow spacethere is no more resistance, no more material to rip. Therefore,discontinuities to some extent increase the robustness of bone ratherthan increase its tendency for brittle fracture (Currey, 1962). In thecase where a crack enters a discontinuity, the front tends to beblunted, hence reducing the stress concentration factor (i.e. the levelof stress necessary to create a crack) and slowing crack propagation.When a crack is forced to enter a vascular canal, the radius at the tipof the crack becomes larger. Lacunae are probably more likely to act asstress concentrators than canaliculae because of their generallyellipsoidal cross-section and because they are generally oriented normalto the long axis. Stress concentrators are define as entities that raisethe stress concentration factor. However, their much smaller sizeprecludes them from acting as fracture initiators (i.e. causes for thestructure to begin fracturing) until or unless plastic deformation hascreated cracks at the tip. Fractures spread along cement lines andlamellar interfaces.

Additional Applications

The model also can be used with complimentary applications andtechnologies. An example includes, but is not limited, to combinationwith software to model soft tissue (such as the one developed by thecompany Infocus, Sylicon Valley, Calif.). Material analogs of bone canbe obtained by means of 3D printers (see, e.g. the printers manufacturedby Stratasys-3D printing in Eden Prairie, Minn.). Implants anddistraction devices will be manufactured by computer guided robots. See,e.g., Mah and Hatcher, 2000. The present model will provide the model ofthe bone structure to be distracted and during distraction.

The application of modeling to imaging (e.g., clinical MRI and CAT scanx-ray imaging) of human bone offers the prospect of a qualitative leapin the predictability, effectiveness, and convenience of surgical,orthodontic, orthopedic and other medical interventions. Embodiments ofthe model can enable medical professionals, based on patient specificdata, to visualize how bone in various parts of the body will grow andheal in response to medical interventions. First, the current lack ofbone local mechanical properties impairs the comparison between naturaland synthetic bones. Second, the current lack of knowledge of mechanicalproperties, strain and stress distributions throughout the bone, impairsthe research for new synthetic bones to move towards the sameproperties. For instance, the latest synthetic long bone is made out offiber-reinforced glass (Szivek, 2000) of unknown local mechanicalproperties. There are no reports of synthetic porous structures with theinterconnecting pores having the same stiffness and strengthcharacteristic of human trabecular bone. Even the most popular syntheticclosed-cell polyurethane foams (such as Daro, Butler, Wis.) which have astructure that shows similarities with human trabecular bone arehomogeneous in theory and with inhomogeneities difficult to control inpractice. In any event the non-homogeneous hierarchical structure ofhuman bone is not even close to being imitated. Third, the current lackof knowledge of mechanical properties and of strain and stressdistributions throughout the bone, impairs bone reconstruction, bonegrafting, placement of screws, insertion of prostheses.

The invention is applicable to the bones of other vertebrates whose bonestructure somewhat differs from that of humans. For instance, theinvention would give valuable results on the prevention and healing offracture in equine bone. Currently, the micromechanical bone studies ofvertebrates are scarce, often limited to a few small animals, such asmouse, dog, and sheep. Because of that, the results on human bonemicrostructure are erroneously used in studies of vertebrates to whichthey do not apply (Riggs et al., 1993a, 1993b).

The present invention provides a more realistic prediction of themacroscopic bone mechanical properties, strain, and stress distributionthan computer models based on omission of either anisotropy ornon-homogeneity of bone. Moreover, this invention provides morerealistic prediction than purely mathematical models, that is modelsbased on hypotheses, which are not based on experimentation. Theliterature is full of research on bone microstructure, which employspurely mathematical models of osteon behavior (Pidaparti and Burr,1992). Such approach is limited, often unrealistic and does not alwayspredict biological phenomena. The invention is flexible so as to includenew experimental findings of bone structural and mechanical properties.This ensures the invention's realistic characteristic insertion ofprostheses, etc.

The present invention will be better understood by reference to thefollowing Examples, which are provided by way of exemplification and notby way of limitation.

EXAMPLE 1

To produce a model of the present invention, compact bone is subjectedto any method that may produce non-invasive slices of biologicalstructures that are known in the art (i.e., μCT-scan or microcomputerized tomography). Images then are stored in a computer and a3-dimensional-reconstruction is applied using a standard method known inthe art (see e.g. Materialise, XYZ Scientific Applications, Inc.Livermore, Calif.). The high resolution of a μCT-scan (about 30 μm)allows for determination of the outline of osteons, of osteons' vascularcanals and interstitial bone. Also the 3D-reconstruction shows varyingshades of gray, which represent the degree of calcification. Osteons arefilled with structure by means of the two lamellar types (bright andextinct lamellae), which have been previously assembled. The criteria bywhich the lamellar structure is drawn into each osteon follows thedistribution of alternate osteons (Hert, et al. 1994) and thedistribution of dominant collagen fibril directions(Portigliatti-Barbos, 1983, 1984, and 1987; Boyde et al., 1984; AscenziA. et al., 1987a and 1987b; Ascenzi A., 1988; Carando et al., 1989 and1991).

The structure of the osteonic lamellar model consists of a laminatewhose length, width and height correspond to a cylindrical shellcircumference, thickness, and height (FIG. 9 a). The structure of thelamella within the interstitial bone is modeled as a portion of theosteonic lamellar model (FIG. 9 b). The layers are unidirectionalfiber-reinforced laminae (FIG. 9 c) of the same matrix and fibers. Thematrix and fibers, i.e. individual components of the hierarchicalstructure, and not the microstructure as a whole, are each treated ashomogeneous and isotropic.

All fibers, of which there are two types, are assumed to be circular incross-section, randomly distributed in the transverse plane. The fibersof the first type have a diameter of about 800 Å and fibers of thesecond type have a diameter of about 200 Å. The fibers of the first typeare proposed to be perfectly embedded in matrix (they are idealized tohave essentially no gaps between them and do not move relative to eachother). The fibers of the second type are proposed to be perfectlyembedded in matrix only when bone undergoes physiological staticloading. When bone undergoes physiological dynamic loading the fibers ofthe second type are given the option to move with respect to matrix inwhich they are embedded. Such displacement is specified followingexperimentation, such as boundary condition or de-bonding experiments.The experimentation may also dictate further conditions for the relativeposition of the two fiber types. Examples of such experiments arediscussed below.

In the present model, the lamina with fiber inclination γ is namedγ-lamina. The thickness of the dark lamella ranges between 4 and 12 μm(Ascenzi M.-G. et al., 2003). It is described by the sequence [82, −82](Frasca et al., 1977). The notation [82, −82] refers to two γ-laminaewhere γ=82, −82. The thickness of the bright lamella ranges between 2and 7 μm (Ascenzi A., et al., 2000). It is described by the sequence[−61.5, −41, −20.5, 0, 20.5, 41, 61.5] (Ascenzi M.-G., 1999b).

For the matrix, Young's modulus of 114 GPa, Poisson's ratio of 0.27, andultimate strength of 0.59 GPa are assumed for hydroxyapatite (Katz andUkraincik, 1971). For the fibers of the first type, Young's modulus of1.2 GPa, Poisson's ratio of 0.35, yield strength of 0.002 GPa are usedfor collagen (Currey, 1969). For the fibers of the second type, Young'smodulus of 1.1 GPa, Poisson's ratio of 0.23, are used formucopolisaccarydes (Boume, 1971). Depending on the degree ofcalcification, the matrix occupies up to 40% of the lamina volumewithout voids (Bonfield and Li, 1967). The cement line is modeled ashomogeneous and isotropic: the Young's modulus of 70 GPa and Poisson'sratio of 0.27 (Philipson, 1965; Schaffler et al., 1987).

Since osteons vary with respect to the distribution of dark and brightlamellae, the model of an osteon with a specific distribution of darkand bright lamellae is obtained by assembling the model of dark andbright lamellae so as to follow the osteon's particular distribution.For instance, a model of the longitudinal osteon, which consists of darklamellae, is made of 12 laminae. The fiber inclination angle changesfrom 82° and −82° six consecutive times. A model of the alternateosteon, which consists of alternating dark and bright lamellae, is madeof 36 laminae. The fiber inclination angle increases by 20.5° from −82°to 82° and then decreases by 20.5° from 82° and −82° four consecutivetimes.

Information included for the present model may not be currentlyavailable for all bones that are evaluated. Any information that isneeded for the practice of this invention may be obtained byexperimentation using methods that are standard in the art.Additionally, methods that may be used to evaluate bone in one speciesmay be used to evaluate a similar bone structure in another species. Forexample, the distribution of dominant collagen bundle directions isavailable for the shaft of the human long bone but not for othervertebrates nor for the mandible. For any compact bone the distributionof dominant collagen fibril directions can be obtained by applying themethod of Boyde et al. (1984).

For cancellous bone the same method can be applied after embedding(soaking and letting dry) the bone in a conventional resin used for thespecimens examined under the electron microscope. Such resin should notchange the microscopic characteristics (birefringence) of the specimens,so that the image of the collagen bundle and hydroxyapatite needledirections under the polarizing microscope is not altered by artifacts.Examples of such resins include, but are not limited to epoxy. Note thatapplication of the invention to cancellous bone will model lamellae thatform trabeculae, as compared with the osteons of compact bone, howevertrabeculae and osteons can both be modeled in terms of lamellae.

From the above-mentioned mechanical properties of matrix and fibers(e.g., Young's modulus and Poisson's ratio) the same types of mechanicalproperties for lamellae under various load types (such as tension,compression, shear, and torsion) will be deduced by means of standardfiber-reinforced laminate methods known in the art (see e.g., Jones,1975; Vinson, 1993; Antman, 1995).

Based on mechanical properties of the lamellae, homogenization theorywill allow for the deduction of the osteon, osteon group, andinterstitial bone mechanical properties for compact bone and trabecularmechanical properties for cancellous bone. The mathematically computedmechanical properties of lamellae, osteons, osteon groups, interstitialbone, and trabeculae are compared to the experimental results. If theexperimental results are not available for the particular bone to whichthe invention needs to be applied, the properties may be determinedusing the methods for mechanical testing as described herein. Themechanical properties of lamellae, osteons, osteon groups, interstitialbone, and trabeculae are used as input for the homogenization methods todeduce the mechanical properties of the desired macroscopic bone.

Results are included of a finite element model, which allows for theassessment of the mechanical properties of the sample. The sampledimensions before and after testing allow for the formulation of anequation that describes the deformation from the shape before testing tothe shape after testing. The deformation equation allows for thecomputation of the strain distribution throughout the sample. Forexample equations see Antman, 1995. The combination of such straindistribution with the experimental diagrams, the known sample structurebefore testing, and the fracture patterns after testing allow for thecomputation of the elastic properties through standard finite elementmethods. Statistical student t-test (Moore and McCabe, 1989) is runacross the sample's results to allow for comparison of mechanicalproperties across the samples and to allow statistical conclusions.

These studies provide the mechanical properties of all the hierarchicalorders. Therefore, the mechanical property distribution throughout thebone in terms of the microstructural components is known. The finiteelement method is applied (see e.g. the software package Abaqus) tocompute the bone response to any given force acting on it. Boundaryconditions are entered as assumptions into the finite element method.The first step is to create a 3-dimensional mesh (see e.g. Couteau etal., 2000).

The bone overall shape is filled with “elements”. These elements areused to represent the osteons present in the bone. For example, a hollowcylindrical portion of an osteon with an inner and diameter of 40 μm, anouter diameter of 220 μm, and height of 500 μm, is filled with about600,000 elements. Mechanical properties and boundary conditions are themethod's input. Boundary conditions express the movements of theboundaries of the various hierarchical structures under loading. Forexample, dynamic loading evidences bone's viscous behavior. Theliterature points to mucopolysaccarides or perhaps collagen as themicrostructural component responsible for the viscosity. In thestructural part of the invention the second type of fibers models themucopolysaccarides. The fibers of the second type are free to move. Suchmovement at the interface between the fibers of the second type and thematrix is expressed by a boundary condition (to be determinedexperimentally). Another example, the behavior of the cement line underloading is the boundary conditions for the osteon and the interstitialbone between which it lies. If the boundary conditions of a specificbone, to which one wants to apply the invention, are not available inthe literature; they can be assessed experimentally by applying methodsdescribed herein or that are well known in the art. The softwareapplication gives as output the strain and stress distributionsthroughout the bone.

The mechanical properties of compact bone microstructure (lamellae,single osteons, osteon groups, single trabeculae) can all beexperimentally found with the following method (other acceptablemethods, including non-invasive methods are available in theliterature). Human cadaveric bone aged between 20-50 is obtainedaccording to the US regulations. The cadaveric bones are chosen eitherfree of pathology to apply the invention to normal bone or with aspecific pathology to apply the invention to a specific pathology. Thebone marrow is removed by standard anatomical techniques(Wickramasinghe, 1975). At least 15 samples of any of such structures(lamellae, single osteons, osteon groups, single trabeculae) areisolated from the surrounding bone. The samples have about the same sizeand shape. The shape is a parallelepiped, a cylinder, or a hollowcylinder (depending on the chosen structure) with lugs (see e.g. FIG. 4c) for mechanical testing. Sample preparation and selection of compactbone microstructure is achieved by the methods of Ascenzi A. et al.(1994, 2000). For example, although any technique can be used, themethod of Ascenzi A. et al. (1994) is preferably used to isolateosteons. The preferred form chosen for isolation of osteon samples is acylindrical shape around the vascular canal. In general, the shape andlocation of a structural sample are chosen in such a way so that all theproperties of the structure are preserved. Mechanical testing of osteonsamples (Ascenzi M.-G. et al., 2000) may include, but is not limited to,monotonic and cyclic testing in tension, compression, shear, bending andtorsion. The methods conducted as described in Ascenzi A. and Bonucci,1967; Ascenzi A. and Bonucci, 1968, 1972; Ascenzi A. et al., 1990, 1994;Ascenzi A. et al., 1985, 1997, 1998 have proven themselves successful.The testing is conducted within the elastic range and beyond the elasticrange to study fractures. Sample preparation and selection of cancellousbone (single trabecula and trabecular groups) is achieved by any of themethods whose bibliography is listed in Mente, 2000. Each sample ismeasured (the three dimensions for the parallelepiped; base radius andheight for the cylinder base inner and outer radii and height for thehollow cylinder) before and after isolation and before mechanical test.Change in dimensions before and after isolation and before mechanicaltest shows existence of prestress. The structure of the sample isassessed before or after the mechanical testing (Ascenzi M.-G. et al.,2000). The samples are tested mechanically under physiologicalconditions, that is wet at 21° C. Since both compact and cancellous boneare viscoelastic, the results of mechanical testing are time-dependent(Sasaki, 2000). Consequently the strain rate and testing frequency needto be prechosen and the computer modeling depends on such choices. Thestress-strain experimental curves (either monotonic or cyclic) throughthe elastic and plastic ranges are evaluated and recorded. After themechanical test, the bone samples are measured and observed under theoptical microscope for fracture patterns.

A trapezoid is cut around each osteon sample (see FIG. 13). For osteonimmobilization during lamellar isolation, a portion of the bone materialinside the trapezoid around the osteon is glued with Kemi®Cyak adhesiveto a slide. The bright and the extinct lamella at the periphery of eachosteon are dissected with a razor-sharp microscopic blade, obtained byfiling a steel needle. To avoid fracture formation during straighteningof each lamellar sample, such operation is performed gently on wetsamples while checking under an optical microscope. The selection ofexternal lamellae, of lesser curvature than internal lamellae, decreasesthe risk of fracture formation during flattening. The ends of eachflattened sample are secured to two supports. The samples are measuredas previously described and examined under an optical microscope toassess defects.

The mechanical loading on lamella on wet samples is conducted staticallyat 21° C. to complete rupture, with current model of the microwaveextensimeter. Flattened bright lamellar sample is expected to resisttension along its length better than extinct lamellar sample. Indeed,the bright lamella is hypothesized to contain collagen bundlestransverse to the longitudinal lamellar axis when enclosed in bone.Therefore, the enclosed lamella's transverse bundles strengthen theflattened lamellar sample in the direction of its length. The extinctlamella is hypothesized to contain collagen bundles parallel to thelongitudinal lamellar axis when enclosed in bone. Hence, the enclosedlamella's longitudinal bundles after flattening are a source of sampleweakness in the direction of its length because they are transverse tothe loading direction. Fracture patterns of ruptured samples are studiedunder an optical microscope. Observation of fractures in rupturedsamples will allow formulation of hypotheses on fracture nucleation andgrowth.

Since the interest is to test the isolated and flatten lamellar samplesmechanically, the stresses present in the flatten lamellar samplesbefore testing are assessed. To compute the stresses in the wetflattened lamellar samples, a computerized geometric-material model of abright wet lamellar sample and of an extinct wet lamellar sample will beconstructed, separately before and after isolation and flattening. Thebright lamella includes prestress. It is hypothesized that the thestresses in the flat bright lamella are larger than the ones in theextinct lamella. Additionally, by taking into account that theperiosteous is prestressed in tension, it may very well be that theouter circumferential system is prestressed in tension, too.

The geometry of the model is based on (1) dimensions (inner and outerradii, height, and dimension variations) of wet lamellar samples beforeisolation from surrounding alternate osteon and after isolation andflattening (width and length, and dimension variations) and (2)structure of lamellar sample. Therefore, dimensional measurements areneeded. The structure model also is based on the lamellar structuralcomponents' arrangement. Therefore, lamellar structure under a confocalmicroscope will be assessed.

EXAMPLE 2

A. Sample Preparation, Measurements, and Experimentation

Sample preparation and selection in this example will be achieved byapplying the Ascenzi A. et al. (1994) methodology. Accurate samplepreparation is important, and the Ascenzi method is preferred. Thefemoral shafts of human corpses free from evident skeletal faults willprovide the bone material for the study. Longitudinal shaft segmentsabout 30 mm long will be first sawn off, and longitudinal sectionsslightly thicker than an osteon (350 μm) will then be prepared using aLeitz rotating-saw microtome. A continuous water-spout will beincorporated to prevent any overheating of the material. Osteon sampleswill then be isolated from the sections. The features determining sampleselection will be the degree of calcification and the orientation of thecollagen bundles and crystallites. Microradiographic examination,preferably according to Amprino and Engström (1952), will allow theselection of fully calcified osteon samples. Two types of osteons willbe selected. They correspond to two different collagen bundle patternsin fiber orientation in successive lamellae. Under the polarizingmicroscope, one type, the longitudinal osteon, has a predominantly darkappearance in cross section (FIG. 4 a); whereas the other type, thealternate osteon, reveals alternately bright and dark lamellae (FIG. 4b).

When bone sections are cut longitudinally, the two osteon types are easyto recognize provided that the thickness of the sections is much lessthan the diameter of an osteon. Longitudinal osteons appear to be almostuniformly bright under the polarizing microscope, while alternateosteons show alternatively bright and dark lamellae. When the thicknessof the bone section differs little from the mean diameter of the osteon,concentric lamellae overlap, thereby reducing or precluding thevisibility of dark lamellae, and leaving open the possibility that analternate osteon may have a bright appearance. As a result,identification only becomes certain once a cross section has been cutfrom the osteon using a microscopic drill (Ascenzi A. et al., 1994).Hence, cyclic loading must be performed before undertaking positiveidentification of the osteon type.

With reference to the position and orientation of the haversian canal,it is necessary that the canal lie midway between the surface of thecylindrical sample and parallel to it, so that torsion is applied aroundthe osteon axis. This calls for the preliminary separation of the osteonsample, e.g. by application of a technique described below. Thistechnique allows the position and orientation of the canal to becalculated by measuring its distance from the outer surface of thesample at various levels and rotational angles.

The samples are isolated in two stages. During the first stage thesample, consisting of the central portion of an osteon, 500 μm inlength, with the ends penetrating into two rectangular lugs, isseparated from the bone section using a device as described in AscenziA. and Bonucci (1968) and Ascenzi A. (1990). As isolation of the centralportion of the osteon is achieved by drilling, its section has a coarse,square shape. During the second stage, a micro-grinding lathe is used togive the central portion a cylindrical form, with the haversian canalrunning through it axially. The lathe to be used was designed anddeveloped by the CECOM Company and is described by Ascenzi A. et al.(1994). The device grinds the sample by a minute steel blade whose edge,500 μm long, is equal to the length of a coarsely isolated sample. Theforward and backward movements of the blade are monitored by amicrometer. The length and other dimensions of the various samples werekept virtually constant; one criterion for the choice of the samples isthat their haversian canal measures 40±3 μm in diameter. Additionally, astopper controls the forward and backward movement of the steel blade onthe micro-grinding lathe to provide a series of samples whose externaldiameter equals 210±3 μm. This provides a precise comparison of samples'torsional properties (Ascenzi A. et al., 1990). Osteons are not uniformin dimensions. With the dimensions carefully controlled and standardizedto exclude defects and other structures, the material rather thanstructural properties are determined for the osteons. This informationcan then be applied to osteonal structures of varying dimensions underthe assumption of homogeneity at the level of the osteon rather than forthe macroscopic specimen.

The relative dimensions of the osteon samples may not appear to conformto those conventionally suggested for material testing. They reflectconditions made necessary by the distinctive nature of bonemicrostructure. In particular, 500 μm is the maximum length compatiblewith the avoidance of Volkmann's canals in the wall of the specimen. Anexternal diameter of 210 μm is the maximum dimension possible thatensures that portions of the neighboring structures are not included inthe sample as a result of irregularities in the thickness of an osteon.The internal diameter of fully calcified osteons averages 40 μm.

FIG. 4 c shows a completely isolated osteon sample held withinrectangular lugs. The lugs allow the sample to be firmly attached to thedevice while hysteretic loops are recorded. The central portion of eachsample will be only 500 μm long; consequently, the sample will notinclude Volkmann's canals which would behave as discontinuities. Inaddition, the osteon sample selection criteria includes that thevascular canal should run strictly parallel to and equidistant from thesurface of the cylindrical sample and that there should not be smallsurface defects. The canal's position and orientation are assessed bychecking the distance between vascular canal and external surface ofsample at various rotational angles and levels. To exclude the presenceof small surface defects that could alter the shear modulus values intorsional testing, each sample is subjected to careful opticalmicroscope examination. Severe criteria are set for osteon sampleselection. Osteon types can only be identified from a prepared crosssection only after a sample has been tested. This means that to obtain60 samples divided between those containing longitudinal and alternateosteons, which satisfactorily complete the procedures adopted for therecording of the hysteretic loops under torsion, it will be necessary toprepare between 800 and 1,000 samples.

The apparatus is an adaptation of the device for testing osteons undertorsion to failure described in Ascenzi A. et al., 1994), and furtherdescribed in FIGS. 10 and 11. This device consists of a rotational axis,point (1) in FIG. 11, with two sets of jaws, point (2) in FIG. 11, whichgrip the specimen during testing. The jaws are oriented along the sameaxis but none of them are free to move axially. This sets up an axialloading effect, which could influence absolute measurements but may beneglected when, as in this investigation, comparative measurements areconsidered.

One set of jaws is fixed, while the other turns in synchrony with awheel, point (4) in FIG. 11, measuring 61 mm in diameter. In order tominimize the rotating friction of the turning jaw, a pendulum loadingsystem is adopted. The axis of the pendulum loading system is indicatedas point (5) in FIG. 11. The frictionless fulcrum of the pendulumloading system is the tip of a hard metal wedge, point (3) in FIG. 11.The maximum oscillation of the pendulum is fixed at 55°. A tungstenthread, whose section measures 20 μm in diameter, winds around the rimof the wheel as a series of 0.1 gram weights is attached incrementallyat one end of the tungsten thread. Weights are added one by one untilfailure occurs for monotonic torsional loading. The load limit is chosenso that the corresponding torque is approximately equal to the middlevalue between the maximum possible elastic torque and fully plastictorque. Preliminary trials indicate a load limit of 0.9-1.0 gram forfully calcified longitudinal osteons and of 0.8-0.9 gram from fullycalcified alternate osteons and of 0.6-0.7 gram for decalcifiedlongitudinal osteons and of 0.5-0.6 gram from decalcified alternateosteons. Once the load limit is reached, the weights will be detachedone by one. The procedure will then be repeated at the other end of thetungsten thread. In this way, the osteon specimen rigidly clamped at oneend is progressively twisted at the other end by a torque incounterclockwise and clockwise directions alternatively, so as toachieving cyclic loading. The interval between the application of twoconsecutive weights will be kept constant at 4 sec. A stereoscopicmicroscope will be used to verify that the axis around which torsionoccurs coincides with the osteon axis. The aim of this operation is tocheck that the center of each jaw corresponds to one end of the osteonsample canal. The angle through which one end of the specimen twistsrelative to the other during testing is measured by applying an opticalmethod based on the reflection of a laser beam from a small mirrorattached to the rotating set of jaws. The variations in the angle oftwist are read on a graduated scale placed 160 cm from the device. Theprecision and accuracy of the graduated scale coincide with those of theapparatus, as checked by applying experimental procedures. Because thediagrams obtained when testing begins in the counterclockwise (orpositive) direction should look essentially like the diagrams obtainedwhen testing begins in the clockwise (or negative) direction, all thediagrams will be recorded starting in the counterclockwise direction,according to the standard practice reported in the literature.

The cycles applied to each specimen will vary in number and they will beinterrupted before the spontaneous sample rupture. Preliminary specimentesting taken to rupture gives an indication of the angle-of-twistvalues at rupture and therefore give an indication to stop cycling whenangle-of-twist values get close to the preliminary rupture values. Thefinal cycle will therefore be interpreted as the cycle preceding theinterruption of the experiment; in consequence, it has no physicalmeaning. Issues related to osteon fatigue-life will not be part of thisstudy because they require the ultimate destruction of the specimenwhich would prevent the ability to properly identify the sample aftercyclic loading. Interruption of the experiment prior to rupture isnecessary because, as previously noted, the osteon type can only beidentified with certainty by preparation of a cross-section beforerupture, but after testing. Before preparation of the cross-section foridentification, samples will be examined under the optical microscope toanalyze the nature and size of lesions. Osteon samples are examinedunder an optical microscope as whole, even though to do so samples needto be removed from the torsion device; such removal can cause structuralchanges, e.g. partial closure of cracks. The nature of fatigue-damagecannot be verified directly under microscope. In fact, fractures due tocycling and those due to osteon sawing to produce the section are notdistinguishable. There is no universally accepted technique available toallow qualitative observation of fatigue-damage. The only experimentalalternative is x-ray diffraction, which provides only quantitativeindication of fatigue-damage (Ascenzi A. et al., 1998).

Of 60 osteon samples, 7 longitudinal and 7 alternate osteon samples willbe decalcified by treatment in a versene solution buffered to pH 7(Ascenzi A. et al., 1967). Measuring the increase in birefringence atregular intervals of time will check the decalcification (Ascenzi A. andBonucci, 1964). After hydration of the material with saline solution, 20longitudinal and 20 alternate fully calcified osteon samples, and the 7longitudinal and 7 alternate decalcified osteon samples will be testedwet as described above. The remaining 6 osteons will be tested on thefirst and second cycles only and their dimensions will be measured (bymeans of a micrometer under an optical microscope) after cycling andbefore the sectional cutting for osteon type determination. Themechanical testing will be conducted at 20° C. The specimens will bemaintained wet during testing by continuous use of a micropipette.

Upon completion of the experimental portion of the research, hystereticvalues of torque vs. angle of twist will be plotted for each osteonsample.

The proposed mechanical testing is performed quasi-statically so ratedependencies not expected in the material response. To confirm this, aseries of twist and hold experiments will be performed as a preliminarytest to see if significant strain relaxation (creep) occurs in the timeframe that it takes to complete an experimental cycle. The presence ofcreep would manifest itself by maintenance of the shape of the curve butwith a clear translation of the entire hysteresis loop. If creep is notpresent, the area under the hysteresis loop is the energy absorption. Ifcreep is present, it will be accounted for by the addition ofviscoelastic terms.

B. Mathematical Analysis of Hysteretic Curves

To establish analogies and differences among plots, the characteristicsof torque vs. angle of twist plots are quantified by means of apolynomial approximation for each half-cycle. (Shiga et al., 1970;Ascenzi A. et al., 1997).

Measurements and Plots

(a) Least-square regression will be applied to the data for eachhalf-cycle to identify the best polynomial approximation of at leastsecond order. The degree cannot be 1 because the half-cycle involves theinelastic range which is characteristically nonlinear.

(b) The goodness of the approximation will be determined throughanalysis of the residuals and computation of the percent variation intorque explained by the regression. A lower bound for r2 is set to 0.98.The degree of the polynomial approximation at (a) will be increased in astepwise fashion until a good-approximation is found for allhalf-cycles.

(c) Let n be the smallest integer for which there is agood-approximating polynomial for all half-cycles. Let the polynomialequation be y(x)=Σa_(i)x^(i). Note that such an equation does notinclude symbolism that denotes the half-cycle; this is done to addclarity and does not lead to confusion. A literature search suggeststhat n might equal 3 or 4 (Shiga et al., 1970 and Ascenzi et al.,1997a). Each coefficient a_(i) for any given cycle will be plotted withrespect to the maximum angle-of-twist on that cycle, for negative andpositive torque half-cycles separately for each osteon type, tovisualize the value's variation as the number of cycles increases.

(d) The first derivative of the polynomial, y′(x)=Σia_(i)x^((i-1)), willbe taken to represent the stiffness of each individual half cycle.

(e) Compute relative extrema of y′(x) on each half-cycle. Because ofincreasing structural damage, stiffness is generally either alwaysdecreasing on both negative and positive torque half-cycles or it showsa sharp change from decreasing to increasing only once on eachhalf-cycle. Therefore, y′(x) should show no relative maximum and amarked relative minimum. If such relative minimum exists, the graph ofy(x) shows an inflection point, i.e. pinching is present. Let h_(m) andk_(m) denote twist and torque values at the stiffness relative minimumof the half-cycle. In these notations, y′(h_(m)) denotes the minimumvalue of stiffness.

(f) Compute y′(x_(b)) and y′(x_(e)), the stiffness at the begin and atthe end of each cycle.

(g) Compute the absolute maximum of the function |y″(x)|/(1+(y′(x))2)3/2on any given half-cycle. This is the maximum curvature per half-cycleand will be denoted by mc.

(h) Compute the difference between the integral of y(x) over the firsthalf-cycle and the integral of y(x) over the second half-cycle. Suchvalue approximates the area of the region bounded by the first cycle.This is the energy absorption during the first cycle and will be denotedel.

Differences in hysteretic behavior between longitudinal and alternateosteons will be established by studying the distributions of maximumtwist, h_(m), k_(m), y′(h_(m)), y′(x_(b))−y′(x_(e)), m_(c), and elcomputed above. The statistical t-test, paired or unpaired, will beapplied on means of distribution or on the mean of the distribution'slogarith_(m) if the distribution is not normal.

The mean of the differences of twist limits between the last and firstcycle obtained from the experimental diagram will be computed. It willbe compared between negative and positive torque half-cycles forlongitudinal and for alternate osteons, separately. The magnitude ofsuch value should be smaller for longitudinal osteons than alternatingosteons because longitudinal osteons resist torsion better thanalternate osteons.

The signs of h_(m) (and k_(m), respectively) at the first and last cyclewill be analyzed. h_(m) (and k_(m), respectively) should have the samesign for the two osteon type separately, up to possibly a few samples.This would indicate that the twist at minimum stiffness should notchange much at all within all negative and positive torque half-cyclesseparately, for both osteon types. This would be in agreement with thetight and well organized osteon structure.

A paired two-sample t-test will be applied to the means of the values ofh_(m) (and k_(m), respectively) at each of the cycles in any given setof corresponding cycles of the two osteon types. This compares thevalues of minimum twist (and torque, respectively) for negative andpositive torque half-cycles.

A paired two-sample t-test will be applied to the means of y′(h_(m)) andthe coefficients of y′(x) for the two osteon types, separately at eachof the cycles in any given set of corresponding cycles of hystereticdiagrams. This compares the minimum value of the stiffness between thenegative and positive torque half-cycles.

An unpaired two-sample t-test will be applied to the means of h_(m) (andk_(m), respectively) for the two osteon types at each of the cycles inany given set of corresponding cycles of hysteretic diagrams. This willcompare the values of twist (and torque, respectively) at the inflectionpoint for the two osteon types.

A unpaired two-sample t-test will be applied to the means ofy′(h_(b))−y′(h_(e)) for the two osteon types at each of the cycles. Thiswill compare the stiffness decrease within a given cycle between the twoosteon types.

A paired t-test will be used on the mean of the difference of m_(c) fornegative and positive torque half-cycles at each of the cycles in anygiven set of corresponding cycles of hysteretic diagrams. This willcompare the maximum value of the curvature of the stress-twist diagrambetween negative and positive torque half-cycles, for the two osteontypes, separately. The maximum value of curvature of the torque-twistdiagram is expected to be larger on the positive than negative torquehalf-cycle for longitudinal osteons; whereas, the maximum values of thecurvature of the torque-twist diagram under negative and positive torquehalf-cycle for alternate osteons should show no difference. As aconsequence of the reduced stiffness the energy absorption should belarger for longitudinal than for alternate osteons.

A paired t-test will be used on the difference of the means of eachcoefficient a_(i) of y′ at last and first cycles, on both negative andpositive torque half-cycles, separately. For both osteon types, thevalue of y′ is expected to decrease from the first to the last cycle onboth negative and positive torque half-cycles for any value of the twistx. This test will measure stiffness degradation here defined as thedecreasing of stiffness at any given twist value on either a negative orpositive torque half-cycle as the number of cycles increases.

The existence of the value h_(m) shows the S-shape of the half-cyclesthat identifies pinching. Pinching is expected to be present for eachcycle for both types of osteon. If pinching is present, pinchingdegradation will be computed.

A paired two-sample t-test will be applied to the means of y′(h_(m)) forthe two half-cycles of any given cycle. This compares pinchingdegradation for the two osteon types separately. Pinching degradation atany cycle is the reduction in stiffness from its value at the deflectionpoint of the negative torque half-cycle to a lesser value at thedeflection point of the positive torque half-cycle.

An unpaired two-sample t-test on the means of the value of minimumstiffness y′(h_(m)) for longitudinal and alternate osteons at any givenhalf cycle. This compares pinching degradation between the two osteontypes.

An unpaired two-sample t-test will be applied to the means of el forlongitudinal and alternate osteons. This compares energy absorptionbetween the two osteon types at each of the cycles in any given set ofcorresponding cycles of hysteretic diagrams.

The mechanical meaning of some of the parameters used in the aboveanalysis of experimental diagrams (e.g. stiffness, energy absorption)will be made clear as such parameters will be correlated toultrastructural behavior during fracture propagation in the modeldescribed at section IV. Up to this point they are comparative measuresof behavior between longitudinal and alternate osteons under torsionalloading.

C. Interpretation

A structural and biological interpretation of the shape of torsionalhysteretic loops in osteons through the results of the previous stepsuses a segment representation of each cycle of the curvilinear recordeddiagram.

If, as it is anticipated, pinching exists, the bilinear model of FIG. 12is appropriate. Here points B and E approximate endpoints of thenegative torque half-cycle, while points E and H approximate endpointsof the positive torque half-cycle; segments DC and FG approximatetangent lines to the curves at the inflection points. The three segmentsmodeling the twist decreasing branch show that stiffness decreases(along segments BC and CU) to a minimum value and then increases (alongsegments UD and DE). Similarly, the three segments modeling the positivetorque half-cycle show that stiffness decreases (along segments EF andFL) to a minimum value and then increases (along segments LG and GH).

The slope of segment DC on the negative (FG on the positive,respectively) torque side of the bilinear model is smaller than both theslopes of segments ED and CB (EF and GH, respectively) and thereforeresponsible for a contraction of the cycle that constitutes pinching.The existence of pinching resides in the torque-angle-of-twist branch ABof the primary curve, where lesions appear as a result of yielding ofcomponents of the bone matrix under load as the angle-of twist increasesto the right. Reversal of loading is required to close the lesions; thiswill occur once the minimum angle-of-twist of the unloading portion BCof the curve is exceeded. Once point C is passed, stiffness, as negativetorque is decreasing, shows a progressive, slight, unsteady increase topoint D. The lesions are then repaired and stiffness rises steadily topoint E.

The opposite will occur at the diagram portions marked EF, FG, and GH.After passing the minimum angle-of-twist of the unloading portion (EF),progressive resolution of the damaged structural components will occur,leading to a slight, unsteady increase in stiffness as negative torqueincreases (FG). After point G, stiffness will increase steadily to pointH, as negative torque increases. In particular, pinching wouldcorrespond to segment CD on the negative torque side, where repair ofthe lesions occurs, and to segment FG on the positive torque side, whereresolution of the lesions occurs. This explanation, in which lesionsform on the negative torque half-cycle and reinforcements yield on thepositive torque half-cycle, does not take into account buckling. Ifbuckling occurs, the situation is reversed: lesions form on the positivetorque half-cycle and reinforcements yield on the negative torquehalf-cycle.

If pinching does not exist, there is no contraction along the cycle. Inthis case, the slope of segment DC on the negative (FG on the positive,respectively) torque side of the model lies in between the values of theslopes of ED and CB (EF and GH, respectively), as depicted in FIG. 12 b.The slope of the segments modeling the cycle shows that stiffnessdecreases along all of them; which might mean that lesions do not repairand do not resolve as they would with pinching. In this scenario it isreasonable to assume that torsional loading creates lesions distributedin a more disorderly fashion in the osteon than in tensile-compressiveloading (aligned with longitudinal bundles and where pinching ispresent). Before load reversal, the hydroxyapatite crystallites mighthave detached and cracked in a way that the original alignment isdestroyed. In this case a partial realignment does not occur later alongthe first half-cycle as to bring an increase in stiffness. Consequently,there are no lesions to resolve during the successive half-cycle andstiffness would keep decreasing. This explanation, in which lesions formon the negative torque half-cycle and reinforcements yield on thepositive torque half-cycle, does not take into account buckling. Ifbuckling occurs, the situation is reversed. Before load reversal,collagen bundles might yield in a way that loses track of the originalalignment. Similarly this does not allow even a partial realignment totake place later along the first half-cycle, after lesions involving thecracking of hydroxyapatite have occurred, so as to increase stiffness.Consequently, no partial alignment of bundles could be restored beforebundles start yielding during the successive half-cycle and stiffnesswould keep decreasing. Such interpretation, where lack of hydroxyapatitecrystallite alignment is greater under torsional cyclic loading thantension-compression cyclic loading, could be verified additionally bymeans of X-ray diffraction (Ascenzi A. et al., 1998).

The results of the measurements and plots above will give the positionsand inclinations of the segments in FIG. 12. For instance, the t-test onthe means of k_(m) along the first half-cycle for longitudinal andalternate osteons is expected to indicate that k_(m) is significantlyhigher for alternate osteon. This means that point U in FIG. 12 a ishigher for alternate osteons, that is, cracks close and stiffnessswitches start increasing for a smaller torque value. This is consistentwith the more complex structure of alternate osteons.

D. Fracture Model

The purpose of the fracture model is to show that cumulativemicro-cracking, de-bonding, void growth and fiber breakage associatedwith repeated loading of osteons causes a progressive loss of stiffnessand pinching, and increase of energy absorption. The lesions observedunder an optical microscope in osteon samples subjected to cyclictorsional loading will serve to develop osteon models and to formulatebiological hypotheses on propagation of fractures. The fracture modelwill be based on:

-   -   hypotheses on ultrastructural components' behavior under cyclic        torsional loading formulated from the experimental hysteretic        plots;    -   ultrastructural components' mechanical properties;    -   fractures observed in osteons during monotonic torsional        loading; and    -   fractures observed in macroscopic bone specimens.

This aspect of the model is an adaptation and extension of the approachof Gupta and Bergstrom (1998). The fracture propagation model is amicromechanical bone model that allows prediction of the progressivegrowth of faulting zones, by considering the increased stressexperienced in the vicinity of an already highly cracked region. Thenucleation of initial damage is determined by the assessment of thepoints more susceptible to fracture. The progressive growth of the faultnucleus is considered in a statistical manner by the use of stressenhancement factors, which address the increased probability of failurein the vicinity of regions that are already cracked.

The geometric model of each of the longitudinal and alternate osteonsamples before mechanical testing consists of a hollow cylinder withcoaxial lateral surfaces. Its internal diameter, external diameter, andheight equal 40 μm, 210 μm, and 500 μm, respectively. Each such hollowcylinder presents voids, and about 20% of each such hollow cylinderconsists of voids (Piekarski, 1970) which model canaliculae and lacunae.

The material model of each of the longitudinal and alternate osteonsamples before mechanical testing consists of a laminate whose length,width, and height correspond to cylindrical shell circumference,thickness, and height, respectively (FIG. 9 a). The layers areunidirectional fiber-reinforced laminae (FIG. 9 b) of the same matrixand fibers. The matrix and fibers are each treated as homogeneous andisotropic. The fibers are assumed to be circular in cross-section with adiameter of 800 Å, randomly distributed in the transverse plane andperfectly embedded in matrix. The lamina with fiber inclination γ isnamed γ-lamina. The elastic properties of matrix and fibers model theelastic properties of hydroxyapatite (Katz and Ukraincik, 1971) andcollagen (Currey, 1969). The matrix occupies 40% of the lamina volumewithout voids (Bonfield and Li, 1967). The matrix (fiber, respectively)volume decreases (increases, respectively) slightly from inner to outerlamina (Amprino and Engstrom, 1952; Rho et al., 1999).

The longitudinal osteon model consists of 9 longitudinal lamellae of thesame thickness. Longitudinal lamellae are modeled by alternating+82-laminae and −82-laminae (Frasca et al., 1977). The alternate osteonmodel consists of 7 transverse lamellae with 5 longitudinal lamellaelayered between them (Giraude-Gille, 1988). The transverse lamella ismodeled by the laminar sequence [−61.5, −41, −20.5, 0, 20.5, 41, 61.5](Ascenzi M.-G., 1999b). This sequence is subjected to prestress asdescribed in Ascenzi M.-G., 1998a and 1999b. A longitudinal lamellarmodel is 9.45 μm thick and a transverse lamellar model is 5.40 μm thick(e.g. Gebhardt, 1906; Ziv et al., 1996). The matrix volume is 10% higherin the longitudinal lamellar model than it is in the the transverselamellar model (Marotti et al., 1994).

E. Bone Structure Simulation

To model fracture propagation in osteons, each of the longitudinal andalternate osteon models is divided into a discrete number of elements,e.g. 618,317. The element mesh will be refined to achieve a convergentsolution. A computer simulation program, such as a Monte Carlosimulation, will be used to do the following tasks.

-   -   1. For any given value of torque applied to experimental        samples, the distribution of stress in the osteon model is        computed. Such computation will take into account voids.    -   2. Such stress distribution will be added to the distribution of        prestress.    -   3. The strain associated to the resulting stress will be        computed on each phase within each element.    -   4. From the strain associated to the resulting stress, the        overall deformation of the hollow cylindrical shell will be        computed.    -   5. From the strain in each phase within each element, the phase        deformation will be computed.    -   6. The strain in each phase within each element is compared to        the yield strain.    -   7. The strain is chosen as the criterion for osteon failure        (Piekarski, 1970). The maximum strain, called critical strain,        after which fracture occurs within each phase is provided by the        literature. Perfect bonding at the interface between phases is        assumed unless experimentally observed cracks appear to initiate        at this interface. If that is the case, a failure criterion        (e.g. Von Mises) will be included.    -   8. The elastic properties of fractured phases will be computed        by means of formulas of the type        E_(i)=E_(e)/(1+(1+ν_(e))(k_(e)λ)/2) (Gupta and Bergstrom, 1998).    -   9. The element is declared broken if all phases in that element        have failed.    -   10. The elements are assumed to be aligned in independent rows        such that the problem of fracture propagation becomes one        dimensional.    -   11. To model the progressive growth of damage, the torque will        be increased incrementally, and using the fracture criterion        above, the number of failure elements will be established.    -   12. The increased probability of fracture in the neighborhood of        an already fractured element will be considered using the        concept of stress enhancement factors.    -   13. If all elements on one row are broken, the strain level at        which all elements on one row are broken is taken as the failure        strain for that row. The process is repeated for each row in the        model. Once the maximum torque is reached, the program stops.    -   14. At this point, the simulation of fracture propagation before        the first hysteretic half-cycle is completed.    -   15. The program incrementally applies a clockwise torque        decreasingly to the maximum torque applied experimentally to        samples, and at each increment it repeats steps 2 and 3 above,        so as to complete the failure simulation during the first        half-cycle of the first cycle.    -   16. Step 15 is repeated for the corresponding counterclockwise        torque so as to complete the fracture simulation during the        second half cycle of the first cycle.    -   17. The fractures obtained according to the model should be        comparable with those observed in osteon samples submitted to        one cycle only.    -   18. The fracture simulation sequence is repeated through the        last cycle.    -   19. Fracture simulation is repeated further, as desired.        Fractures obtained in this way, according to the model, should        be compatible with those observed in cycled osteon samples at        the last cycle.

F. Results

Entities computed from the experimental hysteretic diagrams, such asstiffness degradation, pinching degradation and increase in energyabsorption, up to the second cycle, correlate with the fracturepropagation of the fracture model.

The same geometric/material models and computer program will be used tosimulate fracture propagation under tension, compression, and shear,separately. The resulting fractures will match the fractures observed inosteon propagation under tension, compression, and shear respectively(Ascenzi A. and Bonucci, 1967 and 1968; Ascenzi A. et al., 1972).

Predictions and phenomena simulated according to the model of theinvention include that in both osteon types, a fracture starts at aweaker point of the bone structure (Carter et al., 1981), at the weakinterfaces between two outer lamellae (e.g. Piekarski, 1970), presumablybecause of the hydroxyapatite decrease in osteons from vascular canal toouter wall (Rho et al., 1999).

In longitudinal osteons, the fracture starts somewhat longitudinally,between collagen bundles. It then deviates once or twice at thebeginning of the fracture and is soon followed by a smooth crackadvancing rapidly across the osteon to possibly end in the vascularcanal. As cycling continues, collagen bundles between cracks break, andcracks join to create one or more long almost vertical cracks.

In alternate osteons, cracks are expected to spread obliquely byfollowing the weak interfaces of lamellae. The transverse and obliquecollagen bundles may break before the longitudinal ones as the osteonsection enlarges. Cracks spread through lamellae less rapidly than inlongitudinal osteons as explained by the crack propagation control,characteristic of composite materials (Cook and Gordon, 1964). Once thecrack breaks through transverse and oblique bundles, it will propagatefaster straight through the vascular canal. A long crack should show anoblique orientation between upper and lower extremities.

It follows, unexpectedly, that the longitudinal osteon is weaker inlongitudinal than in transverse shearing, while expectedly the alternateosteon is weaker in tension than in shear (Ascenzi et al., 1967 and1972). This is because when a torque is applied to a body, tensile andcompressive stresses are produced on the lateral surface and torsionalshearing stresses are produced on the cross-section of the body. Thetensile and compressive stresses act approximately at a 45° angle to thelong axis of the body. The direction of the shearing stress on thecross-section of the body is the same as that of the force producingtorsion. If a material is weaker in longitudinal than in transverseshearing, the first cracks arise from axial shearing stresses and appearin a longitudinal direction. However, if the material is weaker intension than in shear, it usually cracks along a spiral course inclinedat a 45° angle to the long axis of the body. The reason for this is thata state of pure shear is equivalent to a state of tension in onedirection and of compression in the opposite direction (Timoshenko andYoung, 1940). The tension stress produces a spiral crack in the body.

For both osteon types 3 to 4 small cracks form in the hydroxyapatite andcollagen, which yields and pulls and/or buckles and makes the cracksspread within lamellae. Microcracks form ahead of the advancing fractureline. Afterwards, during torque reversal, width of cracks anddeformation decrease. Collagen may buckle and some resisting strengthmay appear at zero torque. As cycling continues, cracks extend throughthe lamellae and join.

The slow propagation of cracks in the areas containing transverse andoblique collagen bundles allows for the area to absorb a large amount ofenergy. Slow propagation is essentially a pull-out type mechanism, thatis, hydroxyapatite crystallites are pulled out of the collagen by shearfailure at the fiber-matrix interface. The rapid propagation of cracksin areas containing approximately vertical collagen bundles allows verylow energy absorption. This should be compatible with larger areasenclosed by cycles of experimental plots of alternate osteons (see lastt-test in Sec. A above).

Hydroxyapatite crystallites are pulled out from collagen aroundcanaliculae.

At low strain rates in compression distortion of the lamellar structuresoccurs (McElhaney and Byars, 1965).

The propagating crack generally has the tendency to avoiddiscontinuities (Piekarsly, 1970), hence increasing its length.Discontinuities act as crack arresters by blunting the tip of the crackwhich enters them.

The fracture model is expected to agree with fractures observed inosteons cycled only for first and second complete cycle. The dimensionsof the hollow cylindrical model after one cycle, two cycles, and thelast cycle of torsional loading should match the means of the osteonsamples' dimensions measured experimentally. Furthermore, the role ofthe models' fibers is expected to check with the cyclic behavior ofdecalcified osteons.

The sudden shift of the osteon shape (FIG. 4) from a circular to asquare cross-section suggests a stress concentration at the lugs.Therefore, fractures may begin at the end of some samples earlier duringloading than would otherwise be expected.

Lamellar thickness and width were measured on 20 bright and 20 extinctperipheral lamellar samples in quintuplicate in dry osteon samples byDelta Sistemi IAS 2000 image analysis system, and again after wettingwith a micro-pipette. This table shows means and standard deviations.Thinner extinct lamellae were used for comparison with bright lamellae.It is known that extinct lamellae are thicker than bright ones, whetherdry or wet. The Student t-test is run on the data to determinestatistical differences between dry and wet lamellar dimensions.

Thickness Thickness Width Width Sample Dry Wet dry wet Bright 3.30 ±0.88 3.56 ± 0.93 70.30 ± 9.28 72.45 ± 9.58 Extinct 4.13 ± 1.23 4.10 ±1.10 70.30 ± 9.28 72.45 ± 9.58

Whether dry or wet, bright lamellae are significantly thinner thanextinct lamellae when enclosed in alternate osteons (this agrees withprevious results, e.g. J. Y. Rho, P. Zioupos, J. D. Currey and G. M.Pharr (1999) Bone 25: 295-300). Additionally, wet and dry conditionsaffect bright and extinct lamella thickness differently. Bright lamellaeare significantly less thick when dry than wet. In contrast, extinctlamellae thickness remains constant whether wet or dry.

The bright lamella thickness increases from dry to wet which may be dueto the higher quota of mucopolysaccarides, which expand with water, andto the transverse collagen bundles in the bright lamella tightlyencircling extinct lamella, thereby impeding expansion. Height of bothbright and extinct lamellae is significantly smaller when dry. Inaddition, the thickness along lower and upper borders shows variationsup to 50-60%. This will be included in the model. Width variation isvery low.

The model provides an advantageously simplified simulation orrepresentation of osteon structure. For example, partially calcifiedcollagen bundles are excluded from the model. The model provides auseful and improved description of bone structure and mechanics, eventhough the shape and dimensions of hydroxyapatite crystallites and therelationship of these parameters to the organic components of the matrixare only partially known. Not all collagen bundles are completelycalcified. Those, which are note calcified take up crystallites only on400 A bands (Ascenzi, A. et al., 1965). Such bundles may be comprised ofrelative more stiff 400 A bands separated by relatively more flexibledecalcified collagen segments. In a preferred embodiment of theinvention, partially calcified collagen bundles are not modeled, infavor of modeling fibers in uncalcified collagen bundles. The matrix,which models the hydroxyapatite crystals, lies outside the fibers.Pinching is incorporated into the model is related to the yielding andbucking of fibers, and provides an approximation of the yielding andbuckling of partially calcified collagen bundles. In preferredembodiments, fracture propagation is modeled and cracks will tend topropagate before buckling is likely to occur, because the model in mostcases assumes that individual fibers are perfectly bonded to and areuniformly supported by the matrix. The model also excludes complexconsideration of pore fluids in preferred embodiments which balancerelative simplicity with achieving a reliable and accurate bone model.

EXPERIMENTAL DESIGN OF EXAMPLES 3-6

The experiments conducted as described in Examples 3-6 examine thebehavior of elementary components of adult human compact bone tissueduring dynamic loading of its microstructure. The link betweenbiological composition and mechanical function are investigated throughseveral sets of experiments. For example, one set addresses propertiesof the ultrastructure, and another addresses the behavior of themicrostructure under dynamic torsion.

In particular, these experiments examine secondary osteons to evaluatewhether collagen bundle arrangements and degree of calcificationindicate or determine the relative percentages of collagen andmucopolysaccharides and characteristics of dynamic loading behavior.“Secondary osteons” are present in adult bone, as opposed to “primaryosteons”, which are present at earlier stages of development. Theseexperimental data can be used to further create the numerical model forbone viscoelastic behavior according to the invention.

The present invention also provides experimental designs to assembledata for the model of the invention. For example, osteon samples ofvarying hydroxyapatite crystallite density and collagen bundlearrangements can be mechanically tested under dynamic torsional loadingat low and high strain rates. The viscoelastic properties identified canbe related to the biochemically obtained relative percentages ofcollagen and mucopolysaccharides. A computerized geometric-materialmodel, based on the experimental findings and observations of samplefracture, is then created to describe viscoelastic osteon behavior undertorsional loading in terms of microcracking, debonding, breakage, andvoid growth.

The model of the invention is founded on the recognized dependency ofbone's macrostructural properties on osteon properties established byosteon ultrastructure (Rauber, 1873; Evans, 1958; Currey, 1959; Evansand Vincentelli, 1969; Vincentelli and Evans, 1971; Portigliatti-Barbos,1983, 1984, and 1987; Boyde et al., 1984; Ascenzi A. et al., 1987a and1987b; Katz, and Meunier, 1987; Ascenzi A., 1988; Carando et al., 1989and 1991; Boyde and Riggs, 1990; Crolet et al., 1993; Hert et al., 1994;Ascenzi M.-G., 1999a). The model is also based on observed mechanicalproperties of osteons, using methods of the invention, which can neitherbe inferred from macroscopic samples nor derived by purely mathematicalmodels.

Indeed, some microstructural properties are absent at themacrostructural level and are not susceptible to inference by purelymathematical models (Pidaparti and Burr, 1992). For instance, osteonsamples under monotonic torsional loading display a shear modulus largerthan that of macroscopic samples. This unanticipated result arisesthrough localized slippage at cement lines in larger macroscopicsamples. Also, tension-compression hysteretic loops of osteon samples(Ascenzi A. et al., 1985, 1997) are S-shaped, in contrast to themonotonically increasing behavior of macroscopic specimens. This resultwas not anticipated due to incomplete knowledge of the ultrastructure aswell as the absence of such phenomena in macrostructural samples,possibly due to the mechanical role of cement lines. Close linkage ofmathematical models to micromechanical empirical results such as thesehas led to prediction of lamellar stiffness (Ascenzi M.-G., 1999b)subsequently confirmed empirically (Meunier, 1999; Rho et al., 1999).

Examples 3-6 will elucidate three specific elements towards thelong-term goal of understanding how human bone macro-structure respondsto function. These elements are as follows.

Viscoelastic Properties of Macroscopic Compact Bone Need Clarification(Sasaki, 2000). Starting in 1965 with Currey, and with Smith and Keiper,viscoelasticity has long been recognized as one of the importantproperties of bone. However, divergent experimental conditions andparameters impede meaningful comparison of the various time-dependenttensional and compressional studies on macroscopic compact bone samples,e.g. Smith and Keiper, 1965; McElhaney, 1966; Lugassy, 1968; Sammarco,1971; Black, 1972. Comparison under the hypothesis of linear-viscosity(Lakes and Katz, 1974) found conflicting results and pointed to amacroscopic nonlinear-viscoelastic behavior. This conflict may arisefrom the lack of consideration of the heterogeneity of the macrosamples'ultrastructural composition, i.e. collagen bundles and hydroxyapatitedensities and their arrangements. Time-dependent torsional studies onmacroscopic samples show nonlinear-viscoelastic behavior (Lakes andKatz, 1979a, b, c). For osteon groups, Frasca and Katz's studies suggesta decreasing trend of strain values at the onset of plasticity withincreasing number of osteons. The authors explain such decreasing trendin terms of mucopolysaccharides, the principal constituent of cementlines. None of the above-mentioned results derives from a systematicstudy or has been experimentally correlated to the samples' micro- andultra-structural components. The present work relates the osteonviscoelastic properties to the properties of the ultrastructuralcomponents. While the mineral phase behaves purely elastically (Katz andUkraincik, 1971), collagen and mucopolysaccharides are viscoelastic innature. The known non-linear viscoelastic properties of collagen arelimited to studies from tendon (Haut, 1983). There are no explicit dataavailable on the viscoelastic properties of mucopolysaccharides frombone. According to the invention, the viscoelastic behavior of bonemicrostructure depends on the viscoelastic behavior of collagen andmucopolysaccharides and the structural organization of collagen bundledirections. Mucopolysaccharides are believed to be bonded to thesurfaces of collagen and/or hydroxyapatite crystals and embedded withinthe ground substance (Herring, 1971; Butler, 1984).

Dynamic Torsional Loading, Used to Evidence the Viscoelastic Propertiesof the Macrostructure, Serves to Incorporate the Role of Microstructurein Terms of Ultrastructure. Torsional loading in bone has been analyzedusing finite element analyses (e.g. Martens et al., 1980; Spears et al.,2001). These models suffer from oversimplifications that positnon-existent structural symmetries and homogeneities. Such assumptionsdo not reflect the heterogeneity of the micro level. Macroscopic samplesdo not always have the same mechanical properties as the microstructurethat comprises them, and overlooking them impedes a realisticunderstanding of bone mechanics. For instance, experiments onquasi-static torsional loading of microstructures revealed (Ascenzi A.et al., 1994) that the torsional shear moduli of osteons are much largerthan shear moduli obtained for macroscopic samples. That is, slendersamples are stiffer than thick ones. Slippage of osteons at the cementlines during torsion of macrosamples may explain the lower stiffness inthick samples. Cosserat elasticity theory (Lakes, 1995) well describessuch slippage because it allows a moment per unit area in addition tothe usual force per unit area of classic elasticity theory.

Mechanisms for Initiation and Propagation of Viscoelastic Fracture ofMacroscopic Bone. Dynamic fracture propagation has not been modeled interms of ultrastructural components, although the fracture mechanism ofbone depends on bone structural and composition properties such ascollagen architecture and collagen content (e.g., Jepsen et al., 1999).Jepsen finds that fracture in macroscopic samples is ductile and thatfracture alters bone's viscous mechanism; in particular, relaxationincreases with the increasing extent of fractures. Consequently, viscouseffects in osteon samples can increase with the increasing extent offracture.

According to the invention, and as described in the Examples, osteonsamples show a linear viscoelastic behavior, at least in thephysiological strain range, explained by the Ramberg-Osgood equation.Osteons consisting of longitudinal collagen bundles (longitudinalosteons) resist torsional stresses better than osteons consisting ofalternatively longitudinal, oblique, and transverse collagen fibers(alternate osteons) regardless of the relative hydroxyapatitecrystallites percentage. The viscoelastic effects are less evident inlongitudinal than alternate osteons because longitudinal osteons containrelatively less collagen and mucopolysaccharides, two viscosity factors.The viscoelastic effects are more evident in both osteon types at lowerrather than higher levels of hydroxyapatite crystallites because lowerhydroxyapatite levels correspond to higher mucopolysaccharides levels.

Elementary Components of Secondary Osteons

Knowledge of compact bone microstructure results from a variety ofrefined techniques and sophisticated mechanical devices focused on thebehavior of the complex biological and mechanical interplay ofultrastructural components in the microstructure as a whole (for areview, see Ascenzi M.-G. et al., 2000). Such research has shown, forexample, how collagen and hydroxyapatite distribution and densitydetermine the mechanical properties of osteon and lamellar specimens.

The studies described in the Examples relative to the ultrastructure ofosteons' lamellae and the viscous behavior of single osteons provide anovel perspective on the link between viscoelasticity and osteonultrastructural parameters of collagen and hydroxyapatite crystallites.Through these experiments, the viscous behavior of the single osteon issystematically evidenced for the first time in isolated osteon specimensof the same shape and dimensions. The results confirm the non-systematicfindings of Frasca et al. (1981) on single osteons of variousdimensions.

Various authors, e.g. Katz and Ukraincik (1971) and Frasca et al.(1981), point to collagen and mucopolysaccharides as the componentsresponsible for viscosity, because they are viscous in nature eventhough the known non-linear viscoelastic properties of collagen arelimited to studies from tendon (Haut, 1983). However, up until now,explicit data have not been available on the viscoelastic properties ofmucopolysaccharides from bone. Osteonic lamellae show molecularorganization of a mostly collagen bundle organic framework andhydroxyapatite crystallites of orientation analogous to that of thecollagen bundles. Mucopolysaccharides are embedded in the groundsubstance and are thought to be bonded to the collagen and/orhydroxyapatite crystal surface (e.g. Herring, 1971; Butler, 1984). Whenthe ground substance is removed from several connective tissues, thetime-dependent mechanical effects decrease (Minns et al., 1973). Thisresult further points to the role of collagen and mucopolysaccharides inthe viscous behavior of osteons. Further, the relative percentages ofmucopolysaccharides and hydroxyapatite in single osteons are indicatedto be inversely proportional (Pugliarello et al., 1970). In fact, themucopolysaccharides need to decrease for the formation of a substrate onwhich the hydroxyapatite crystallites can deposit and continue toaccumulate (Herring, 1971; Sasaki and Yoshikawa, 1993).

The present invention considers lamellar structure in these examples andin examples 1, 2, and 7-14. Thus, the invention addresses the longstanding question of whether the histological and radiologicaldifferences observed in osteon lamellae are due to differences in fiberorientation (as proposed by Ebner, 1887 and elaborated by Gebhardt,1906) or composition (as theorized by Ranvier, 1887; Ruth, 1947;Rouillier et al., 1952; Rouillier, 1956). Such theories concernparticularly the osteon type consisting of an alternating series of twodifferent lamellar types (so-called alternate osteon). While Gebhardt'stheory has for a long time served as a useful model in approaching andclarifying the micromechanical behavior of isolated osteonic specimens,at present Gebhardt's theory is far from universally accepted. Theexperiments described in the Examples support a combination of the twotypes of theories. On one hand, these findings support a version ofGebhardt's model, previously supported by various authors e.g. Frasca etal. (1977); Giraud-Guille (1988); Ascenzi A. et al. (1982). That is,dark and bright lamellae differ in collagen bundle and hydroxyapatitecrystallites orientation. In particular, lamellae which appear dark incross-section under polarizing light, contain longitudinal collagenbundles and hydroxyapatite patterns. Lamellae which appear bright incross-section under polarizing light contain transverse collagen bundlesand hydroxyapatite patterns. On the other hand, the same new techniquesindicate differences in composition. As described in the Examples,longitudinal collagen bundles lie closer to each other than oblique andtransverse collagen bundles, leaving less space between them for theground substance and therefore mucopolysaccharides. A strikingdifference is found between the hydroxyapatite pattern orientationwithin the two lamellar types. Lamellae which appear dark incross-section under polarizing light are thicker than lamellae appearingbright, confirming the observations of Marotti (1993). Further, wetlamellar thickness varies in relation to collagen bundle orientation andthat the different changes in lamellar thickness between dry and wetstates point to a difference in mucopolysaccaride percentages andcollagen bundle orientation in the two lamellar types.

This shows that the viscoelastic behavior of single osteons depends onthe viscosity of collagen and mucopolysaccharides, whose relativepercentages depend on collagen bundle directional arrangement and theamount of hydroxyapatite present.

Osteon Specimens

Osteons vary in diameter (200-300 μm) and length (up to a fewcentimeters). Osteons consist of generally coaxial cylindrical layerscalled lamellae that are only a few microns in thickness. Osteons varyin terms of arrangements of dark and bright lamellae in cross sectionunder (circularly) polarized light and in degree of calcification fromdark gray to white on micro-x-ray. For a meaningful systematic study,all osteon specimens need to have the same dimensions and structuralcharacteristics while maintaining the characteristics of the grossentity from which they are selected as specimens. The fourabove-mentioned variables can be dealt with realistically as follows:

Shape and dimension of specimens. A central cylindrical portion of theosteon around a 40±3 μm vascular canal with an external diameter of210±3 μm avoids the irregularities of interstitial bone and a length of500 μm avoids the discontinuities of Volkmann's canals.

Collagen bundle make-up. The examination of compact bone sections underthe polarizing microscope reveals that all osteons are composed oflamellae that appear dark and lamellae that appear bright. Examinationof thousands of bone sections has revealed that osteons made upexclusively with bright lamellae are rare; that all prevalently darkosteons show a thin layer of bright lamellae around the harvesian canal;and that among all the different combinations of dark and brightlamellae in osteons, osteons made up of essentially dark lamellae(so-called longitudinal osteons) and osteons made up of alternativelydark and bright lamellae (so-called alternate osteons) represent twoends of a spectrum, biologically and mechanically. Therefore,longitudinal and alternate osteons were chosen for the investigationsset forth in the Examples.

Degree of calcification. Examination of micro-X-rays of compact bonesections shows a spectrum of shades, from dark gray to white, whichindicate the degree of calcification from initial to final (Amprino andEngström, 1952) of osteons, as a whole. Osteons at the initial stage ofcalcification comprise 5-10% of adult compact bone, and this percentagedecreases with age. Osteons at the final stage of calcificationconstitute the majority, close to perhaps 90%. The level ofhydroxyapatite in osteons at final stages of calcification is usuallyconsidered to remain constant within a 27 to 49 age group. In apreferred embodiment, osteons at initial and final stages ofcalcification obtained from donors in this age group are modeled by theinvention.

Number and thickness of lamellae. The distribution of lamellar numberand thickness varies from osteon to osteon. Lamellar thickness fromlamella to lamella from the cement line to the haversian canal (Rho etal., 1999; Ardizzoni, 2001) and within the lamella along the osteonlength (Ascenzi A. et al., 1982) also varies from osteon to osteon.Lamellar number can be counted and thickness measured at both ends ofthe cylindrical portion of the osteon specimens under polarized light.Since the lamellar boundary is wavy along the lamellar 500 μm length,the thickness values obtained under polarized light for dark (bright,respectively) lamellae may be smaller (larger, respectively) than theones obtained on thinner (70-100 μm) sections. While this potentialdiscrepancy can affect absolute measurements, it does not affectcomparative investigations. About 15-20 osteon specimens, quitehomogeneous with respect to lamellar number and thickness of lamellae,can be obtained from the mid-diaphysis of a single human femur.

EXAMPLE 3 Osteon Viscous Behavior and Mucopolysaccharide Content

This Example reports two studies which provided information on osteonviscous behavior. The results show that the mucopolysaccharide contentis higher in alternate rather than longitudinal osteons. These osteonscan be modeled accordingly.

The bone material came from the femoral shafts of human cadavers, agedbetween 18 and 55, obtained in accordance with American regulation andfree from evident skeletal faults and infectious diseases such as AIDS,hepatitis A and B, and syphilis.

First Study

Six osteon samples, obtained as described below, were subjected to twistand hold testing to evidence their viscous behavior through relaxation.Osteon sample isolation was achieved by applying the methodologydescribed in Ascenzi A. et al. (1994). By means of a rotating-sawmicrotome, longitudinal segments about 30 mm long were first sawn fromthe femoral shaft. The segments were sliced in 350 μm thick longitudinalsections (i.e. slightly thicker than an osteon). A continuouswater-spout was incorporated to the saw to prevent any overheating ofthe material.

The samples for torsional loading (FIG. 22C) were isolated in two stagesfrom longitudinal sections. During the first stage the sample,consisting of the central portion of an osteon, 500 μm in length, withthe ends penetrating into two rectangular lugs, was separated from thebone section using a specially constructed device which includes adental drill as described in Ascenzi A. and Bonucci (1968) and AscenziA. et al. (1990). As isolation of the central portion of the osteon isachieved by drilling, its section has a coarse, square shape. During thesecond stage, a micro-grinding lathe was used to give the centralportion a cylindrical form, with the haversian canal running through itaxially. The lathe to be used was designed and developed by the CECOMCompany and is described by Ascenzi A. et al. (1994). The device grindsthe sample by a minute steel blade whose edge, 500 μm long, is equal tothe length of a coarsely isolated sample. The forward and backwardmovements of the blade are monitored by a micrometer. The length andother dimensions of the various samples were kept virtually constant;one criterion for the choice of the samples is that the haversian canalmeasure 403 μm in diameter. Additionally, a stopper controls the forwardand backward movement of the steel blade on the micro-grinding lathe toprovide a series of samples whose external diameter equals 210±3 μm. Therelative dimensions of the osteon samples may appear not to conform tothose conventionally suggested for material testing. They reflectconditions made necessary by the distinctive nature of bonemicrostructure. In particular, 500 μm is the maximum length compatiblewith the avoidance of Volkmann's canals in the wall of the specimen. Anexternal diameter of 210 μm is the maximum dimension possible thatensures that portions of the neighboring structures are not included inthe sample as a result of irregularities in the thickness of an osteon.The internal diameter of fully calcified osteons averages 40 μm.

FIG. 22C shows a completely isolated osteon sample held withinrectangular lugs. The lugs allow the sample to be firmly attached to thedevice while experimental diagrams are recorded. The central portion ofeach sample is only 500 μm long; consequently, the sample does notinclude Volkmann's canals, which would behave as discontinuities. Thiswas determined by inspection under an optical microscope, which alsochecks that there are no small surface defects that could alter theshear modulus values in torsional testing. The canal's position andorientation were assessed by checking the distance between vascularcanal and external surface of sample at various rotational angles andlevels. Because osteon types can be identified from a prepared crosssection only after a sample has been tested, between 1,600 and 2,000samples are prepared to obtain 120 working samples (15 osteon samplesper osteon type at initial and final stages of calcification), whichwill satisfactorily complete the procedures adopted for the recording ofthe experimental diagrams under torsion at low and high strain rates.

The apparatus used is the one described by Ascenzi A. et al. (1994) totest osteons under quasi-static torsion to failure. This device consistsof a rotational axis, point (1) in FIG. 23, with two sets of jaws, point(2) in FIG. 23, which grip the sample during testing. The jaws areoriented along the same axis but none of them are free to move axially.This sets up an axial loading effect, which could influence absolutemeasurements but may be neglected when, as in this investigation,comparative measurements are considered. One set of jaws is fixed, whilethe other turns in synchrony with a wheel, point (4) in FIG. 23,measuring 61 mm in diameter. In order to minimize the rotating frictionof the turning jaw, a pendulum counterbalance system is incorporated.The axis of the pendulum loading system is indicated as point (5) inFIG. 23. The frictionless fulcrum of the pendulum loading system is thetip of a hard metal wedge, point (3) in FIG. 23. The maximum oscillationof the pendulum is fixed at 55°. A thin light thread, whose sectionmeasures 20-60 μm in diameter, winds around the rim of the wheel.Weights may be attached to the thread to load the osteon sample. Theangle through which one end of the specimen twists relative to the otherduring testing is measured by applying an optical method based on thereflection of a laser beam from a small mirror attached to the rotatingset of jaws. The variations in the angle of twist are read on agraduated scale placed perpendicularly to the plane of the device, 40 cmhigher. The precision and accuracy of the graduated scale coincide withthose of the apparatus, as checked by applying experimental procedures.Because the diagrams obtained when testing begins in thecounterclockwise (or positive) direction should look essentially likethe diagrams obtained when testing begins in the clockwise (or negative)direction, all the diagrams were recorded starting in thecounterclockwise direction, according to the standard practice reportedin the literature.

This is how the torsimeter was used: Every 11 seconds a 0.1 gram weightwas attached to the end of the nylon thread to a total of eight weights.The number of weights was eight because the diagrams of Ascenzi A. andBenvenuti (1994) indicate that 0.8 gram loading with this devicecorrespond to a torque between elastic limit and ultimate strength. Theangle of twist corresponding to the 8 weights was recorded after 20seconds and again after another 40 minutes had elapsed. All six samplesshowed a non zero angle-of-twist change, i.e. creep. The meanangle-of-twist change observed equals 1.6°.

Second Study

The longitudinal (alternate, respectively) osteon is constituted bylongitudinal (alternatively longitudinal and alternate, respectively)lamellae (FIGS. 22A and 22B). Longitudinal and transverse lamellarsamples were measured dry and wet inside the alternate osteons, wereisolated and observed flat under a confocal microscope. Suchexaminations yield indication of a higher mucopolysaccharide content intransverse rather than longitudinal lamellae and therefore in alternaterather than longitudinal osteons.

Lamellar thickness and width were measured on 20 transverse and 20longitudinal peripheral lamellar samples in quintuplicate in dry osteonsamples by Delta Sistemi IAS 2000 image analysis system, and again afterwetting with a micro-pipette. This table shows means and standarddeviations. Thinner longitudinal lamellae were used for comparison withtransverse lamellae. It is known that longitudinal lamellae are thickerthan transverse ones (see e.g. Marotti, 1994), whether dry or wet. Thestudent t-test is run on the data to determine statistical differencesbetween dry and wet lamellar dimensions.

TABLE 1 Lamellar Thickness Thickness Thickness Width Width SampleDry(μm) Wet (μm) Dry(μm) Wet (μm) Transverse 3.30 ± 0.88 3.56 ± 0.9370.30 ± 9.28 72.45 ± 9.58 Longitudinal 4.13 ± 1.23 4.10 ± 1.10 70.30 ±9.28 72.45 ± 9.58

Whether dry or wet, transverse lamellae are significantly thinner thanlongitudinal lamellae when enclosed in alternate osteons. Additionally,wet and dry conditions affect transverse and longitudinal lamellarthickness differently. Transverse lamellae are significantly less thickwhen dry than wet. In contrast, longitudinal lamellae thickness does notchange significantly whether wet or dry. The transverse lamellarthickness increase from dry to wet supports the hypothesis thattransverse lamellae contain a higher quota of mucopolysaccarides, whichexpand with water, and that the transverse collagen bundles in thetransverse lamella tightly encircling longitudinal lamella impedeexpansion. The width of both lamellar types is significantly smallerwhen dry.

Three longitudinal and 3 transverse lamellar samples were isolated byapplying the methodology described in Ascenzi M.-G. et al. (2003). Therotating-saw microtome was employed to cut longitudinal segments about30 mm long from the femoral shaft. The segments were then sliced in 70±6μm thick transverse sections. The continuous water-spout prevented thematerial from overheating. On the thin transverse section, a trapezoidwas cut around each chosen alternate osteon sample (FIG. 24). For osteonimmobilization during lamellar isolation, a portion of the bone materialinside the trapezoid away from the osteon was glued with Kemi®Cyakadhesive to a slide. The longitudinal and transverse lamellae at theperiphery of each osteon were dissected with a razor-sharp microscopicblade, obtained by filing a steel needle. To avoid fracture formationduring straightening of each lamellar sample, such operation isperformed gently on wet samples while checking under an opticalmicroscope. The selection of external lamellae, of lesser curvature thaninternal lamellae, decreases the risk of fracture formation duringflattening. The structure of the isolated and flattened lamellar sampleswere observed wet under a Leitz confocal microscope. The confocalmicroscope worked well with the natural fluorescence of wet bone.Because the photomultipliers detect light intensity and not color, redis the color applied to the image.

The longitudinal lamellar samples show a regular arrangement of collagenbundles. From one border to the other, the collagen bundles are parallelto the osteon axis. Each dot is the cut radial collagen bundle thatfollows the osteocyte process. On transverse lamellar samples, collagenbundles of only one oblique inclination were evidenced for the firsttime. The transverse lamellar samples show ample areas of groundsubstance between collagen bundles, parallel and oblique to the flatlamellar borders. The larger areas of ground substance suggest a largerquota of mucopolysaccharides and a perhaps lower quota of collagen intransverse rather than longitudinal lamellae. This supports thehypothesis that, at the same degree of calcification, alternate osteonscontain a larger quota of mucopolysaccharides than longitudinal osteons.

EXAMPLE 4 Viscous Osteon Model

This Example shows the experimental basis for, and subsequentmathematical modeling to create, a viscous osteon model. The study canbe described by the following flow chart.

The bone material for this study conforms to the specificationsdescribed in the previous Example. Sample selection follows thespecifications of Ascenzi A. et al. (1994). The features determiningsample selection are their degree of calcification and their collagenbundle and crystallites orientation. Micro-X-ray, the method of Amprinoand Engström (1952), provides for the selection of osteon samples at theinitial and final stages of calcification. The number of osteons atinitial (final, respectively) stages of calcification decreases(increases, respectively) with age. Osteons at initial stages ofcalcification comprise 5-10% of osteons in the above-mentioned bone age.The two osteon types, which are representative of osteon structures withrespect to collagen bundle patterns in fiber orientation in successivelamellae, were selected for this study. On longitudinal bone sections,the two osteon types are easy to recognize only if the section thicknessis much smaller than the diameter of an osteon. Longitudinal osteonsappear to be almost uniformly bright under the polarizing microscope,while alternate osteons show alternatively bright and dark lamellae.When, as in this case, the thickness of the bone section differs littlefrom the mean diameter of the osteon, concentric lamellae overlap,thereby reducing or precluding the visibility of dark lamellae, andleaving open the possibility that an alternate osteon may have a brightappearance. As a result, identification only becomes certain once across section has been cut from the osteon using a microscopic drill(Ascenzi A. et al., 1994). Hence, torsional loading must be performedbefore undertaking positive identification of the osteon type. On the500 μm thick transverse bone sections, longitudinal osteons appear to bealmost uniformly dark (FIG. 22A); whereas the other type, the alternateosteon, reveals alternately bright and dark lamellae (FIG. 22B).

Samples for Mechanical Testing

These samples are isolated as described above. It is necessary toisolate about 1,600-2,000 to obtain about 120 samples suitable fortesting torsion at low and high strain rates (15 osteon samples perosteon type at initial and final stages of calcification).

Samples for Biochemical Analysis

The samples for biochemical analysis are isolated from 500 μm thicktransverse sections using the technique of Ascenzi A. and Bonucci(1968). No lugs are necessary, and the type of each 500 μm long osteonsample can be easily recognized on transverse sections. The device usedfor sample isolation consists of a very thin, carefully sharpened steelneedle inserted off-center in a dental drill. As the drill turns, thetip of the needle describes a circle whose diameter may be adjusted tomatch that of the particular sample diameter. When the rotating axis ofthe needle is perpendicular to bone section surfaces, i.e. coincideswith the osteon axis, the tip of the needle cuts an osteon sample ofcylindrical shape with walls of uniform thickness just inside itslimits. To ensure that this rotating axis is perpendicular to bonesection surfaces, the drill's handle is inserted in a microscope body inplace of the tube and the section firmly secured onto the microscopestage. Coarse microscope adjustment controls the needle's movements intothe bone section. Osteon cutting is controllable by watching theoperation through a stereoscopic microscope. The length and diameter ofeach sample is accurately measured by means of an eyepiece micrometer.This provides a precise comparison of torsional properties. Individualosteons are not uniform in dimensions. With the dimensions carefullycontrolled and standardized to exclude defects and other structures, thematerial properties rather than variable structural properties aredetermined for the osteons. In the long run, this information can beapplied to osteonal structures of varying dimensions under theassumption of homogeneity at the level of the osteon rather than for themacroscopic sample. The criteria of regularity for mechanical loadingrefer to the position and orientation of the haversian canal. It isnecessary that the canal lie midway between the surface of thecylindrical sample and parallel to it, so that torsion is applied aroundthe osteon axis. This calls for the preliminary separation of the osteonsample by application of a technique about to be outlined. Thistechnique allows the position and orientation of the canal to becalculated by measuring its distance from the outer surface of thesample at various levels and rotational angles. After dissection, theharvesian canal is cleaned out by inserting a metal rod slightly (10 μm,approximately) thicker than the harvesian canal. FIG. 25 shows the shapeof samples prepared with the above-described technique. It is necessaryto isolate about 100 osteon samples per osteon type at initial and finalstages of calcification to have sufficient material volume for each ofthe two biochemical analyses.

Mechanical Testing of Wet Osteon Samples

Mechanical testing of wet osteon samples is performed under monotonictorsional loading. Two types of testing are employed. (1) A torsimeteris used to demonstrate creep of the osteons as a function of theirmakeup. (2) Additional specimens are loaded under displacement controlat constant strain rates of 10⁻² and 10 Hz until rupture of the sampleoccurs. These latter measurements corresponds to low and high strainrates.

Torsimeter Tests

The torsimeter is used as follows.

1) Creep Tests

A 0.8 gram weight is attached at one end of a 20 μm thick tungstenthread, applying a static torque to the specimen. Angle of twist isrecorded initially at 5 second intervals followed by longer intervals asthe creep stabilizes. Preliminary work indicated that additional creepwas insignificant after 40 minutes. The samples are maintainedthroughout the testing by a saline drip irrigation. This test yieldsangle of twist versus time for these quasistatic tests. Initially,Kelvin-Voigt and 3-parameter viscoelastic models are used to determinethe elastic and viscous constants for the specimens. For the threeparameter solid model, for example, the constitutive relation assumesthe form:σ+p ₁ σ•=q ₀ ε+q ₁ε•

where σ is the applied stress, σ• is the stress rate, ε is the strainand ε• is the strain rate. The coefficients p₁, q₀, and q₁ representcombinations of the elastic and viscous constants depending on theparticular form of the model employed. These coefficients (and hence theassociates elastic and viscous constants) are obtained directly from theobserved angle of twist versus time diagrams obtained according to themethod described by Hayes W. (1972).

2) Low and High Strain Rate Tests

For this series of tests the torsimeter is modified for attachment to anMTS 612 servohydraulic testing machine. The fixed stock of thetorsimeter incorporates a reaction torque sensor (5 oz-in capacity). Thedrive shaft is modified to incorporate a bowstring arrangement toconvert the linear actuator displacement to angle of twist. The base ofthe torsimeter is rigidly mounted to centralize the actuator axis withthe bowstring arrangement. Ramp loadings of 10⁻² mm/sec and 10 mm/sec isused to apply torque to the specimens. One output of these results is aplot of torque versus angle of twist recorded in real time. In thesetests, the specimens are all taken to failure. According to theinvention, there is a direct dependency of the slope of the torqueversus angle of twist curves on the loading rate. This is supported bypreliminary results demonstrating a significant creep effect. There isalso a dependence of this slope on the actual structural material makeupof the osteon samples.

After the mechanical testing, osteon samples are examined under anoptical microscope under both regular and fluorescent light (Huja etal., 1999) to assess fracture patterns; measured in terms of diameterand height; and assessed in terms of the number of bright and darklamellae.

The experimental results include, for example, graphs of increasingfunctions of the angle-of-twist, which show a concave downward shapestarting at the origin of the reference system, regardless of the osteontype and the degree of calcification. According to the invention, theplotted results shows a model in which the observed curves are steeperand less concave for the high (rather than low) strain rate, regardlessof the osteon type and the degree of calcification. According to themodel, a similar profile applies to the osteon samples at the final(rather than initial) degree of calcification, regardless of the osteontype, for any fixed strain rate. A similar profile again applies tolongitudinal (rather than alternate) osteon samples, regardless of thedegree of calcification, for any fixed strain rate.

Further, the experimental diagrams of fully calcified osteon samples,according to the invention, are steeper and less concave than thequasi-static experimental diagrams obtained by Ascenzi A. et al. (1994)per osteon type.

Biochemical Assessment of Collagen and Mucopolysaccharides

Biochemical assessment of collagen (as hydroxyprolines) andmucopolysaccharides (as hexosamine) is performed on 150 μg amounts oflongitudinal and alternate osteons at initial and final stages ofcalcification. While it is difficult to quantify small quantitiesprecisely, the uncertainty does not affect the comparative conclusionsto be made (Herring, 1972).

Osteon samples are dried to constant weight in a vacuum desiccator overP₂O₅. Since decalcification of osteons is necessary to quantifyhexosamine and hydroxyproline, acid hydrolysis is used. Residual HCl isremoved before the assays are performed. Hydrolysis products areseparated using refined chromatographic techniques as first described byExley (1957).

Hexosamine is determined spectroscopically essentially according to theprocedure of Elson and Morgan, (Exley 1957) and modified by Oguchi etal. (1979). The methodology is refined to eliminate the possibility ofinterference from amino acids and mineral salts (Pugliarello et al.,1970). Hyroxyproline is determined essentially according to theprocedure of Serafini-Cessi and Cessi (1965), as refined by Teerlink etal. (1989).

The biochemical analysis has been performed successfully (Pugliarello etal., 1970) on osteon samples at initial and final stages ofcalcification, regardless of the osteon type. Moro et al. (2000) haveemployed on rat bone a technique refinement that can be applied toosteons. Collagen and mucopolysaccharides percentages are significantlylower in longitudinal rather than alternate osteons. In fact, alternateosteons contain more collagen and mucopolysaccharides than longitudinalosteons at equal degrees of calcification, because alternate osteonscontain transverse lamellae, which are richer than longitudinal lamellaein collagen and mucopolysaccharides. According to the invention, themucopolysaccharides percentage decrease, as the degree of calcificationincreases, is statistically significant in alternate osteons.Conversely, the mucopolysaccharide percentage decrease will not bestatistically significant, as the degree of calcification increases, inlongitudinal osteons. In either case, the mucopolysaccharide percentagedecreases in osteons, regardless of the osteon type, when the degree ofcalcification increases. Here, the resulting collagen andmucopolysaccharides percentages for longitudinal (alternate,respectively) osteon samples is lower (higher, respectively) than thevalues found by Pugliarello et al., 1970) regardless of the osteon type.The means of collagen and mucopolysaccharide percentages are combinedwith the mean number of longitudinal and transverse lamellae inlongitudinal and alternate osteons to yield the mean percent of collagenand mucopolysaccharides within longitudinal and transverse lamellae, atinitial and final stages of calcification. These percentages are notpreviously reported in the literature.

According to the invention, the biochemical analysis shows astatistically significant higher percent of collagen andmucopolysaccharides in alternate rather than in longitudinal osteonsamples at equal stages of calcification; a statistically significanthigher percent of collagen and mucopolysaccharides in transverse ratherthan in longitudinal lamellae at equal stages of calcification; and astatistically significant (not significant, respectively) decreasingamount of collagen and mucopolysaccharides as the degree ofcalcification increases in alternate (longitudinal, respectively) osteontypes.

Mathematical Modeling

Mathematical modeling will consist of analysis of the experimentaldiagrams to establish yield strength, ultimate strength, moduli andconstitutive equations; and the realization of a computerizedgeometric-structural model, which will simulate the behavior of themicroscopic components so as to include fracture propagation duringloading.

Analysis of Experimental Diagrams

The observation of the experimental diagrams assesses the yield strengthand ultimate strength. For each experimental diagram, the constitutiveequation, which relates stress, strain and their time dependencies, isestablished in terms of the Ramberg-Osgood equation:θ=Tc(dθ/dt)^(d) +aT ^(N)(dθ/dt)^(b)

where θ and T denote angle-of-twist and torque respectively; a, b, c, d,N denote constant values that depend on the material properties, with a,b, c≧0 and d≦0.

Such equations have accurately modeled the response set at variousstrain rates of a wide range of tested engineering materials (Rambergand Osgood, 1943). This is because the geometric shape of theexperimental diagrams changes only moderately as the strain rate varies;and the Ramberg-Osgood equation is a simply formulated polynomial whosecoefficients are functions of the strain rate, which describes therelatively simple geometric shape of the experimental diagrams, i.e. anincreasing function graph, that passes through the origin and is concavedown.

The determination of the constants a, b, c, d and N will follow theprocedure used by Hight and Brandeau (1983) for macroscopic compact bonesamples. The Ramberg-Osgood equation is expected to suffice to produce agood fit (measured by an r² of 0.98-0.99) because the osteonviscoelastic behavior is expected to be less complex than that ofmacroscopic samples. If desired, increasingly more complex differentialequations are employed, starting with more complex polynomials andrational functions of T whose coefficients depend on the strain rate.

Linear-viscosity is expected (Frasca et al., 1977) at least forphysiological strains, which for the outer wall of the tibia (as oneexample) are of the order of between 0.0007 and 0.0020, at physiologicalstrain rates, which are of the order of between 0.0135 and 0.5143 Hz. Iflinear-viscosity is present and the Ramberg-Osgood equation provides agood fit, the coefficients a, b, c, N equal 0, 1, 0, 0, respectively.

The approximating function of each experimental diagram will serve tocompute: the viscoelastic modulus as the derivative of the diagramapproximating function at zero strain and the energy absorption capacityas the area under the approximating curve.

Statistical Analysis

The Two Way Analysis of Variance is applied to the means of viscoelasticmodulus, yield strength, ultimate strength and energy absorption withosteon type (longitudinal or alternate) and degree of calcification(initial or final) as factors for each strain rate. If normality islacking, the 2-way ANOVA is applied to the means of the logarithms.Significance is set at 0.05. The Post hoc Student-Newman-Keuls testidentifies the significant factors. This test yields significantdifferences in the following:

-   -   1. Viscoelastic modulus, yield strength, ultimate strength, and        energy absorption capacity should be higher at the final stages        rather than at the initial stages of calcification, regardless        of the osteon type and the strain rates    -   2. Viscoelastic modulus, yield strength, ultimate strength, and        energy absorption capacity should be higher for longitudinal        rather than alternate osteon samples, regardless of the degree        of calcification and the strain rate.    -   3. Viscoelastic modulus, yield strength, ultimate strength, and        energy absorption capacity should be higher for longitudinal        rather than alternate osteon samples, regardless of the degree        of calcification and the strain rate;    -   4. Viscoelastic modulus, yield strength, ultimate strength, and        energy absorption capacity should show a smaller increase in        value for longitudinal rather than alternate osteon samples, as        the strain rate increases, regardless of the degree of        calcification.    -   5. Viscoelastic modulus, yield strength, ultimate strength, and        energy absorption capacity should increase with increasing        strain rate, regardless of the osteon type and the degree of        calcification.

In certain embodiments of the model, if desired, yield strength is notdetermined.

Viscoelastic Osteon Model

The purpose of the viscoelastic osteon model is to relate the mechanicalbehavior of the osteon sample to the behavior of the ultrastructuralcomponents that causes a progressive loss of stiffness. Themicromechanical behavior is described in terms of micro-cracking,de-bonding, void growth and components breakage. The lesions observedunder an optical microscope in osteon samples subjected to dynamictorsional loading at various strain rates serves to develop osteonmodels and to formulate biological hypotheses on propagation offractures. In a preferred embodiment, this longitudinal osteon model isan extension (so as to include components' viscoelastic properties) ofthe elastic model in Ascenzi M.-G. (2000); whereas the alternate osteonmodel is prepared ex novo. The osteon model is based on the experimentaldiagrams' approximating functions, angle-of twist as function of torque;the hypotheses on ultrastructural components' behavior under dynamictorsional loading formulated from the experimental diagrams; theultrastructural components' percents, as obtained from the biochemicalanalysis; and the ultrastructural components' viscoelastic properties.

The geometric model of each of the longitudinal and alternate osteonsamples before mechanical testing consists of a hollow cylinder withcoaxial lateral surfaces. Its internal diameter, external diameter, andheight equal 40 μm, 210 μm, and 500 μm, respectively. Each such hollowcylinder presents pores, as shown in FIG. 26. Pores in the model willinclude the vascular canal, canaliculae and lacunae and equal 20% of thetotal osteon volume (Piekarski, 1970).

The material model of each of the longitudinal and alternate osteonsamples before mechanical testing consists of a laminate whose length,width, and height correspond to cylindrical shell circumference,thickness, and height, respectively (FIG. 27A). The layers areunidirectional fiber-reinforced laminae of the same matrix and fibers.The matrix and fibers are each treated as homogeneous and isotropic. Thematrix is considered as elastoplastic and the fibers asviscoelastoplastic. The fibers are assumed to be circular incross-section and randomly distributed in the transverse plane. Thelamina with fiber inclination γ is named γ-lamina (FIG. 27B). There aretwo types of fibers. The first fiber type, with a diameter of 800 Å,represents collagen. The second fiber type, with a smaller diameter,represents mucopolysaccharides. The matrix occupies up to 40% of thelamina volume without voids (Bonfield and Li, 1967), at the highestdegree of calcification. The relative percentages of matrix at theinitial stages of calcification and of the two fiber types at initialand final stages of calcification, and the diameter of the second typefibers is based on the biochemical analysis.

The elastic properties of the matrix are seen to model the elasticproperties of hydroxyapatite (Katz and Ukraincik, 1971). Theviscoelastic properties of the fibers of the first type are seen tomodel the viscoelastic properties of collagen (Currey, 1959; Haut,1983). The viscoelastic properties of the fibers of the second type areseen to model the unknown viscoelastic properties ofmucopolysaccharides. Little information is available regarding thefluids within the pores incorporated in the microstructure. Initially,the proposed model simplifies or disregards the structural effects ofthe fluid within these pores with assignments of minimal materialproperty values. This form of the model is exercised parametrically toinclude fluid within the pores, with various bulk moduli.

The longitudinal osteon model consists of 12 laminae 7.08 μm thick. Itis described by the sequence (82, −82) repeated 6 times (Frasca et al.,1977). The alternate osteon model consists of 36 laminae. The fiberinclination angle increases by 20.5° from −82o to 82o and then decreasesby 20.5° from 82o and −82o four consecutive times. Here the ±82-laminaeare 7.08 μm thick while the other laminae are 1.01 μm thick. Because thesequence (−61.5, −41, −20.5, 0, 20.5, 41, 61.5) models a 7.08 μm thicktransverse lamella, transverse and longitudinal lamellar models have thesame thickness. Transverse lamella is subjected to a strain associatedwith prestress (Ascenzi and Benvenuti, 1980), as described in AscenziM.-G., 1998a and 1999b. Longitudinal lamellar model is 9.45 μm thick andtransverse lamellar model is 5.40 μm thick (e.g., Gebhardt, 1906;Marotti, 1993; Ascenzi A. et al., 2001). The matrix volume is 10% higherin the longitudinal lamellar model than it is in the transverse lamellarmodel (Marotti et al., 1994).

Fracture Propagation Modeling

To model fracture propagation in osteons, each of the longitudinal andalternate osteon models is divided into a discrete number of elements.The starting number can be 618,137. The element mesh is refined toachieve convergence of the solution. Then a computer program, based onMontecarlo simulation, is written to perform the following tasks:

1. For any given value of torque applied to experimental samples, thedistribution of stress in the osteon model is computed. Such computationwill take into account voids.

2. Such stress distribution will be added to the distribution ofprestress.

3. The strain associated to the resulting stress will be computed oneach phase within each element.

4. From the strain associated to the resulting stress, the overalldeformation of the hollow cylindrical shell will be computed.

5. From the strain in each phase within each element, the phasedeformation will be computed.

6. The strain in each phase within each element is compared to the yieldstrain.

7. The strain is chosen as the criterion for osteon failure (Piekarski,1970). The maximum strain, called critical strain, after which fractureoccurs within each phase, is provided by the literature. A failurecriterion (e.g., Von Mises) will be included if cracks appear toinitiate at the matrix-fiber interface.

8. The elastic properties of fractured phases will be computed by meansof formulas of type E_(i)=E_(e)/(1+(1+ν_(e))(k_(e)λ)/2) (Gupta andBergström, 1998).

9. The elements are assumed to be aligned in independent rows such thatthe problem of fracture propagation becomes one-dimensional.

10. To model the progressive growth of damage, the torque will beincreased incrementally, and using the fracture criterion above, thenumber of failure elements will be established.

11. The increased probability of fracture in the neighborhood of analready fractured element will be considered using the concept of stressenhancement factors.

12. If all elements on one row are broken, the strain level at which allelements on one row are broken is taken as the failure strain for thatrow. The process is repeated for each row in the model. Once the maximumtorque is reached, the program stops.

13. At this point, the simulation of fracture propagation is completed.

14. The fractures obtained in this way in the model should check withthose observed in osteon samples.

Verification of the Model

Entities computed from the experimental diagrams are correlated, such asstiffness degradation and fracture propagation, to verify the osteonmodel. The osteon model (loaded quasi-statically) should behave like theosteon sample (loaded quasi-statically).

The influence of factor pertaining to the following ultrastructuralcomponents and voids are also considered:

Large strains, which may be accommodated by the organic phase,contribute to the dissipation of energy at the front of a propagatingcrack. Crack propagation appears to be arrested in the presence ofcanaliculae and lacunae. In the case where the crack enters adiscontinuity, its front is blunted, hence reducing the stressconcentration factor and slowing crack propagation. When a crack isforced to enter a vascular canal, the radius at the tip of the crackbecomes larger. Lacunae are probably more likely to act as stressconcentrators than canaliculae because of the ellipsoidal cross-sectionand because they are generally oriented normal to the long axis.However, their much smaller size precludes them from acting as fractureinitiators until or unless plastic deformation has created cracks at thetip and thereby extended them to the critical length for spontaneousfracture. It seems unlikely that smaller discontinuities could act asstress concentrators. It may be that discontinuities “to some extentincrease the robustness of bone” (Currey, 1962) rather than increase itstendency for brittle fracture. No cross-lamellar or cross-osteonalcracks are observed in macroscopic samples.

Examination of Model Results

In both osteon types, a fracture starts at a weaker point of the bonestructure (Carter et al., 1981) at the weak interfaces between two outerlamellae (e.g. Piekarski, 1970; Simkin and Robin, 1974). That outerlamellae are involved in the fracture process is tentatively explainedby the hydroxyapatite decrease in osteons from vascular canal to outerwall (Rho et al., 1999).

In longitudinal osteons, the fracture starts somewhat longitudinally,between collagen bundles. It then deviates once or twice at thebeginning of the fracture and is soon followed by a smooth crackadvancing rapidly across the osteon to possibly end in the vascularcanal. As torque increases, collagen bundles between cracks break, andcracks join to create one or more long almost vertical cracks.Deviations causing a dentate profile may be due to the viscoelastic,strain rate sensitive mucopolysaccharides and perhaps collagen. Suchdentate profile should therefore be more evident at low (rather than)high strain rate and at the initial stages of calcification when theosteons are richer in mucopolysaccharides. In alternate osteons, cracksare expected to spread obliquely by following the weak interfaces oflamellae. The transverse and oblique collagen bundles may break beforethe longitudinal ones as the osteon section enlarges. Cracks spreadthrough lamellae less rapidly than in longitudinal osteons as explainedby the crack propagation control, characteristic of composite materials(Cook and Gordon, 1964). Once the crack breaks through transverse andoblique bundles, it will propagate faster straight through the vascularcanal. A long crack should show an oblique orientation between upper andlower extremities. In alternate osteons at initial stages ofcalcification the cracks may start at a higher value of torque andpropagate more slowly than at the final stages of calcification if thepercent of mucopolysaccharides is higher.

It follows that at the same degree of calcification, a longitudinalosteon is weaker in longitudinal than in transverse shearing while analternate osteon is weaker in tension than in shear (confirmed byAscenzi A. et al., 1967 and 1972). This is because when a torque isapplied to a body, tensile and compressive stresses are produced on thelateral surface, and torsional shearing stresses are produced on thecross-section of the body. The tensile and compressive stresses actapproximately at a 45° angle to the long axis of the body. The directionof the shearing stress on the cross-section of the body is the same asthat of the force producing torsion. If a material is weaker inlongitudinal than in transverse shearing, the first cracks arise fromaxial shearing stresses and appear in a longitudinal direction. However,if the material is weaker in tension than in shear, it usually cracksalong a spiral course inclined at a 45° angle to the long axis of thebody. This is because a state of pure shear is equivalent to a state oftension in one direction and of compression in the opposite direction(Timoshenko and Young, 1940). The tension stress produces a spiral crackin the body.

For both osteon types, 3 to 4 small cracks form in the hydroxyapatiteand collagen, which yields and pulls and/or buckles and makes the cracksspread within lamellae. Microcracks form ahead of the advancing fractureline.

The slow propagation of cracks in the areas containing transverse andoblique collagen bundles allows for the area to absorb a large amount ofenergy. Slow propagation is essentially a pull-out type mechanism, thatis, hydroxyapatite crystallites are pulled out of the collagen by shearfailure at the fiber-matrix interface. The rapid propagation of cracksin areas containing approximately vertical collagen bundles allows verylow energy absorption. This should be compatible with larger areas underthe experimental graphs of alternate osteons.

Hydroxyapatite crystallites are pulled out from collagen aroundcanaliculae.

At low strain rates in compression, distortion of the lamellarstructures occurs (McElhaney and Byars, 1965).

The propagating crack generally has the tendency to avoiddiscontinuities (Piekarsky, 1970), hence increasing its length.Discontinuities act as crack arresters by blunting the tip of the crack,which enters them.

The osteon model agrees with fractures observed in osteon samples afterloading. The dimensions of the hollow cylindrical model after dynamictorsional loading matches the means of the osteon samples' dimensionsmeasured experimentally.

The sudden shift of the osteon shape from a circular to a squarecross-section suggests a stress concentration at the lugs. Therefore,fractures can begin at the end of some samples earlier during loadingthan would otherwise be expected.

Advantages of the Model

The model of the invention advantageously simplifies osteon structure,in particular with respect to exclusion of partially calcified collagenbundles. The shape and dimensions of hydroxyapatite crystallites and therelationship of these parameters to the organic components of the matrixare only partially known. Not all the collagen bundles are completelycalcified. Those that are not calcified take up crystallites only on 400Å bands (Ascenzi A. et al., 1965). Hence such bundles may be comprisedof relatively more stiff 400 Å bands separated by relatively moreflexible decalcified collagen segments. This model does not containfibers that model such partially calcified collagen bundles. Here thefiber of the first type model uncalcified collagen bundles, the fiber ofthe second type model the mucopolysaccharides and the matrix, whichmodels the hydroxyapatite crystals, lies outside both fiber types.

EXAMPLE 5 Properties of Osteons Under Dynamic Loading

The experiments reported here examine the properties of osteons underdynamic loading. They focus on viscoelasticity, characterizing first howviscoelasticity depends on collagen bundle direction and hydroxyapatitedensity, and second how the relative percentages of collagen andmucopolysaccarides depend on collagen bundle direction andhydroxyapatite density. These experimental results therefore provideimportant information on the role that the ultrastructural constituentsplay in the viscoelastic behavior of the osteon. By describing thelimits and extent of the role that the osteon ultrastructure plays indetermining osteon viscoelasticity, a better understanding is gained ofhow bone tissue absorbs energy during dynamic loading and of howfractures propagate.

Furthermore, these experiments can show that longitudinal and alternateosteons show a linearly-viscous behavior in the physiological strainrange. Such a showing would be of fundamental importance in theunderstanding of the non-linear viscous behavior hypothesized by Lakesand Katz (1979a and b) on macroscopic bone specimens. In fact, it wouldpoint to the cement lines, which are more ductile than osteons, asresponsible for the non-linear effect.

Thus, the aim of these studies was three-fold: to evidence single osteonviscoelasticity, subject of the described mechanical experiments; toevidence elementary component differences in osteon lamellae, subject ofthe described chemical experiments; and to identify additional osteongeometric and structural variables in order to clarify long standingquestions and provide a meaningful link among the proposed experimentalresults.

For the four experiments described below, the bone material consisted ofadult human femoral shafts (27-49 years old), free from evident skeletalfaults, and removed from cadavers in accordance with US regulations. Theselected age range corresponds to the young age group of Kuo et al.(1998).

Mechanical Evidence of Osteon Viscous Behavior

Osteon specimens were subjected to twist and hold testing. All specimensshowed viscous behavior as evidenced by relaxation. The result points tothe appropriateness of dynamic loading of osteon specimens toinvestigate osteon viscoelasticity.

Methods. The Ascenzi A. et al. (1994) methodology was used to obtain 6osteon specimens. A femoral shaft was first sawn into 30 mm longlongitudinal segments by means of a rotating-saw microtome with acontinuous watering system to provide lubrication and prevent thematerial from overheating. The segments were then sliced intolongitudinal slabs 350 μm thick, i.e. slightly thicker than an osteon.Micro-X-rays of these longitudinal sections were prepared to allowidentification (Amprino and Engström, 1952) of fully calcified osteons,as desired for this study. Both the section and the micro-X-ray werethen scanned for a 500 μm long portion of fully calcified osteon arounda straight haversian canal of the required diameter (in this study,averaging 40±3 μm) that was free of Volkmann's canals. The identifiedspecimen was then isolated in a two-step procedure. During the firststep, a dental drill secured to a microscope stage cut a coarseparallelepiped around the specimen. During the second step, a CECOMCompany micro-grinding lathe machined the 500 μm long central portion ofthe coarse parallelepiped to the shape of a cylinder that contained thehaversian canal. The external diameter of this cylinder is set tomeasure 210±3 μm. After grinding, the concentricity of the canal wasassessed by checking the distance between canal and external cylindricalsurface of the specimen at various rotational angles and levels. Acanals that deviated more than ±10 μm at any point along its length withrespect to the axis of the cylinder was deemed unacceptable for use.From a classical mechanics approach this is sufficient to ensure thatthe specimen's torsional stiffness (rigidity) does not vary by more than5% due simply to structural anomalies.

The relative dimensions of the osteon specimens reflect conditionsdetermined by the distinctive nature of bone microstructure. Inparticular, (i) 500 μm is the maximum length compatible with theavoidance of Volkmann's canals in the wall of the specimen that wouldbehave as discontinuities; and (ii) an external diameter of 210 μmensures that the central portion of the osteon is isolated and thattherefore portions of the neighboring structures are not included in thespecimen to avoid irregularities in the osteon specimen thickness. FIG.28 b shows a completely isolated osteon specimen with lugs, necessary toattach the specimen firmly to the loading device. Inspection under anoptical microscope checks the specimen for small surface defects thatcould alter the shear modulus values in torsional testing.

The axisymmetric cylindrical shape of the osteon specimen lends itselfwell to torsional loading. At the level of a single osteon the specimenunit appears homogeneous and isotropic by observation. To determineelastic constants by mechanical testing methods, many techniques areavailable. The cylindrical nature of the specimens lends itself ideallyto torsional testing. This loading method is used to determine thematerial stiffness (shear modulus in this case) and is not intended tosimulate in vivo loading conditions experienced on the intact boneduring activities of daily life. This method of loading is appropriateand sufficient for the determination of one of the two elastic constantsthat completely define the mechanical behavior of these specimens as awhole, as isotropic specimens. Extensive experience in the applicationof this technique for intact long bone specimens is advantageous.

The apparatus used is described in Ascenzi A. et al. (1994), to testosteon specimens under quasi-static torsional loading to failure.Weights are attached to a thin light thread (FIG. 23) to load the osteonspecimen in torsion. An optical method measures the angle through whichone end of the specimen twists relative to the other during testing.Although the resulting torque vs angle-of-twist diagrams are independentof the direction of loading, all the loadings were initiated in thecounterclockwise direction. Every 11 seconds a 0.1 gram weight wasattached to the end of the thread, up to a total of eight weights. Thetotal number of weights was indicated by the fact that the 0.8 gramloading with this device corresponds to a torque between elastic limitand ultimate strength during quasi-static experiments (Ascenzi A. etal., 1994). For each specimen, the angle of twist corresponding to the 8weights was recorded after 20 seconds and again after another 40 minuteshad elapsed.

Results. All six specimens showed a non zero angle-of-twist change (seetable below) with a mean of 1.6°. A non zero angular deflection pointsto tissue relaxation and reveals the viscous behavior.

TABLE 2 Osteon Angle-of Twist Change Specimen No. 1 2 3 4 5 6 Angular0.38 1.50 0.75 2.42 1.50 2.76 Deflection (degrees)

Thickness Variation within Dark and Bright Lamellae

The thickness of the wet outer lamellae of alternate osteons wasmeasured along their two ends, separately, on thin transverse sections.Dark and bright lamellae show thickness variation between the two ends.The result points to the appropriateness of thickness measurements onosteon specimens for both mechanical and chemical investigations.

Methods. Transverse section thickness was set at 70 μm. Polarizing lightallowed the identification of alternate osteons. Micro-X-ray allowedidentification of required osteon degree of calcification (Amprino andEngström, 1952), set to final in this preliminary study. Lamellarthickness was measured at 5 points on each of the two ends of 86 darkand 66 bright peripheral lamellar specimens embedded in wet alternateosteons by Delta Sistemi IAS 2000 image analysis system. Table 3 showsmeans and standard deviations (in microns).

TABLE 3 Lamellar Thickness Thickness Thickness Thickness Specimen On End1 On End 2 Difference Dark 4.27 ± 1.00 4.85 ± 0.96 0.29 ± 0.26 Bright7.62 ± 1.91 8.58 ± 1.99 0.48 ± 0.47

Differences between bright and dark lamellae were significant (p<0.05).Both dark and bright lamellar thickness varies from lamella to lamellaand within any given lamella along the osteon length as previouslyobserved by Ascenzi A. et al. (1982) on isolated bright lamellae.

Thickness Variation Between Wet and Dry Lamellar Conditions

Dry and wet lamellar thicknesses were measured within the osteonsamples. Whether dry or wet, dark lamellae are thicker than brightlamellae and bright lamellae expand with water to a lesser degree thanthe dark lamellae. The result points to the appropriateness ofbiochemical analysis to quantify the mucopolysaccharides in longitudinaland alternate osteons.

Methods. The rotating-saw microtome described above was employed to cuttransverse sections. A micro-X-ray of each cross section was prepared toallow identification (Amprino and Engström, 1952) of required degree ofosteon calcification, set to final in this study. Lamellar thickness andwidth were measured at 5 points on each of 20 dark and 20 brightperipheral lamellar specimens in dry alternate osteon specimens by usinga Delta Sistemi IAS 2000 image analysis system, and again after wettingwith a micro-pipette. On each lamella, all measurements were takenconsistently on one of the two transverse ends. Only thinner darklamellae were used for comparison with bright lamellae because darklamellae are in general thicker than bright ones (see e.g. Marotti etal., 1994), whether dry or wet. The mean length (±S.D.) of the wholealternate osteon equaled 70.30±9.28 μm and 72.45±9.58 μm, under dry andwet conditions, respectively. The table shows means and standarddeviations (in microns) on which the student t-test was run withsignificance set at 0.05.

TABLE 4 Lamellar Thickness Thickness Thickness Specimen Dry Wet Dark4.13 ± 1.23 4.10 ± 1.10 Bright 3.30 ± 0.88 3.56 ± 0.93

Whether dry or wet, bright lamellae are significantly thinner than darklamellae when enclosed in alternate osteons. Additionally, wet and dryconditions affect bright and dark lamellar thickness differently. Brightlamellae are significantly less thick when dry than wet. In contrast,dark lamellae thickness does not change significantly whether wet ordry. The bright lamellar thickness increase from dry to wet supports thehypothesis that bright lamellae contain a higher quota ofmucopolysaccarides, which tend to expand with moisture, and that thebright collagen bundles (see sections 4.5-4.7) in the bright lamellatightly encircling dark lamella impede expansion. The length of thewhole osteon is significantly less when dry.

Isolation of Dark and Bright Lamellar Specimens

The invention provides a new technique, which allows isolation of bothdark and bright lamellar specimens. Previously, Ascenzi A. et al. (1982)isolated specimens of dark lamellae by a technique that cannot beapplied to bright lamellae. Here, two lamellar types are examined andcompared through and across their flatten cylindrical surface. Lamellarcollagen bundle orientations and hydroxyapatite patterns were observedby means of three different methodologies (Ascenzi M.-G. et al., 2003).These are: circularly polarizing microscopy, confocal microscopy, andthe X-ray diffraction. Confocal microscopy and small-angle X-raydiffraction were employed for the first time on lamellar specimens. Thethree methodologies yield well-matched results even though the obliquecollagen bundles of dark lamellae observed under polarizing light werenot observed by the other two methods.

Method. The rotating-saw microtome described was employed to cutlongitudinal segments approximately 30 mm long from the femoralmid-shaft. The segments were then sliced in thin (70-100 μm) transversesections of desired thickness. A micro-X-ray of each section wasprepared to allow identification (Amprino and Engström, 1952) ofrequired osteon degree of calcification. Fully calcified osteons wereselected for this study. On the section, a trapezoid was cut with thedental drill described above around each chosen alternate osteonspecimen (FIG. 24). For osteon immobilization during lamellar isolation,a portion of the bone material inside the trapezoid away from the osteonwas glued to a slide. The dark and bright lamellae at the periphery ofeach osteon were dissected wet with a razor-sharp microtome blade,obtained by filing a steel needle. Because it is necessary to hold thebone during isolation, the whole lamella with exclusion of a littletract was isolated. In some instances, to improve the microscopicexamination of particular aspects of the collagen layers of the freesurfaces, specimens of both lamellar types were delicately scratchedwith the needle for micro-dissection. To avoid fracture formation duringstraightening of each lamellar specimen, the operation is performedgently on wet specimens while under direct visualization with an opticalmicroscope. The selection of external lamellae, of lesser curvature thaninternal lamellae, decreases the risk of fracture formation duringflattening. The difficulty of applying this free-hand micro-dissectionto obtain regularly dissected and discontinuity-free specimens isevidenced by the fact that approximately only 1 out 3 specimenssuccessfully complete the procedure.

Circularly Polarizing Light Microscopy on Dark and Bright Lamellae

Dark (bright, respectively) specimens are composed of longitudinal(transverse, respectively) and oblique up to ±45° collagen bundleorientations.

Method. 102 dark and 110 bright lamellar specimens were examined with aLaborlux Leitz equipped for polarizing light. A Zeiss laser scanmicroscope equipped with argon ion and helium-neon lasers was used tolocate the structures of interest and to obtain digital images,respectively. These last were archived on a hard disk and subsequentlytransferred to a film. To increase the digital image fluorescence,structures of interest were stained with a diluted solution of eosin.For result verification, a few 1-2 μm thick section were cut at a slightangle with respect to the transverse plane from decalcified boneembedded in paraffin. The lamellae appeared better separated than in thecross-section and consequently better resolved optically. After stainingwith eosin the sections were observed under a fluorescent microscope.

Results. When a wet lamellar specimen is observed through its flattenedcylindrical surface under a polarizing microscope, its features changein relation to the lamellar orientation with respect to the polarizingplanes of the two Nicol's prisms. When the long edge of the darklamellar specimen is oriented at 45° (0°, respectively) with thepolarizing plane, collagen bundles run perpendicularly (obliquely at±45°, respectively) to the long edge of the specimen. When the long edgeof a bright lamellar specimen is oriented at 45° (0°, respectively) withthe polarizing plane, collagen bundles run parallel (obliquely at ±45°,respectively) to the long edge of the specimen (FIGS. 31 and 32). Thedistribution of the collagen bundles appears homogeneously distributedexcept at some points of the oblique bundle distribution. Suchdiscontinuities may really exist or be due to artificial removal ofbundles during dissection or be due to optical elision of superimposedorthogonal bundles. Collagen bundles reveal intermediate orientations atintermediate orientations of the lamellar specimen with the polarizingplane.

Confocal Microscopy on Dark and Bright Lamellae

Dark (bright, respectively) specimens are composed of longitudinal(transverse and oblique at approximately ±45°, respectively) collagenbundle orientations. Collagen bundles that follow the osteocyte processare observed to run across layers of longitudinal collagen bundles.Confocal microscopy allowed observation of collagen orientations withoutoverlap and of the ground substance between unidirectional areas ofcollagen bundles. Observation of collagen bundle direction and proximitypoints to the appropriateness of a biochemical analysis to quantifycollagen and mucopolysaccharides in longitudinal and alternate osteons.Observations of collagen bundle direction provide information on theosteon structure. According to the invention, oblique collagen bundlesof dark lamellae are observed by confocal microscopy or by polarizingmicroscopy.

Methods. 3 dark and 3 bright lamellar specimens were isolated asdescribed in section 4.4 from nominal 70 μm thick transverse section.Square areas of up to 50×50 μm were scanned wet every 1 μm in thethickness direction by a Leitz confocal microscope. Since the naturalfluorescence of wet bone worked well with the confocal microscope, nostaining of the specimen was necessary. The photomultipliers detectlight intensity and not color; red was chosen to be applied to theimage.

Results. The dark lamellar specimens show a regular arrangement ofcollagen bundles (FIG. 31). From one border to the other, the collagenbundles are parallel to the osteon axis. Each dot is the cut radialcollagen bundle that follows the osteocyte process. On bright lamellarspecimens, transverse and oblique collagen bundles were evidenced withmore space between bundles than in dark lamellae (FIG. 32). Such findingsuggests a lower concentration of collagen. The confocal microscopeallows discrimination of unidirectional areas of collagen bundles ofvarious directions. This observation appears to be new, not elsewherereported in the literature. The observation of collagen longitudinal andtransverse bundle directions confirms Gebhardt's model. The brightlamellar specimens show ample areas of ground substance between collagenbundles, parallel and oblique to the flat lamellar borders. The largerareas of ground substance suggest a higher concentration ofmucopolysaccharides in bright rather than dark lamellae. These findingssupport the hypothesis that, at the same degree of calcification,alternate osteons contain a greater concentration of mucopolysaccharidesthan longitudinal osteons. Whether the low amount (less than 1%) ofmucopolysaccharides will show a significant difference between osteontypes at the same degree of calcification, as predicted remains to bedetermined, e.g., through observation of dark lamellae by confocalmicroscopy.

X-Ray Diffraction on Dark and Bright Lamellae

Lamellar specimens which appear dark and bright, respectively, incross-sections were investigated by Small- and Wide-Angle X-raydiffraction (SAXS and WAXS, respectively) for collagen andhydroxyapatite crystallite pattern orientations. Collagen bundleorientations and hydroxyapatite patterns differ between dark and brightlamellae and follow analogous patterns within the same lamellar type.These results further support Gebhardt's model. Such observationsprovide information on the osteon structural variables underlying thecomputerized osteon specimen model.

Methods. Thirteen fully calcified dark and 13 bright lamellar specimenswere isolated and flattened, as described above, from 70 μm thicktransverse sections. The dental drill with water cooling allowedpreparation of 2 radial hemisections and 2 transverse sections. Thisinvestigation was designed to check the hypothesis of collagen andhydroxyapatite pattern differences between dark and bright lamellae.SAXS and WAXS lend themselves well to such studies because SAXS isindicative of collagen orientation through the “staining” of thecollagen bundles by the hydroxyapatite crystallites while the WAXS isindicative of hydroxyapatite pattern orientation (but not theorientation of single crystals). The direction of the incident beam waschosen parallel (at 45°, respectively) to the lamellar width in dark(bright, respectively) lamellae to investigate patterns parallel(oblique at approximately ±45°, respectively) to the osteon axis (FIG.33).

SAXS and WAXS diffraction patterns were recorded using the scanningdiffractometry setup of the ID13 microfocus beamline of the EuropeanSynchrotron Radiation Facility (Grenoble, France). The beam wavelengthof 0.964 Å was obtained with a Si(111) monochromator and focused to 7 μm(full-width at half-maximum) by a glass capillary. The sample wasscanned through the beam by a computer controlled x/y gantry. Intactscales, as well as portions of scales were mounted on a goniometrichead, with the surface of the scale orthogonal to the X-ray beam. Squareareas of up to 50×50 μm were scanned with spatial resolutions of 5 μmand 10 μm. Diffraction patterns were recorded while moving the samplealong the horizontal and vertical axes. Also, some horizontal andvertical linear scans were performed with spatial resolutions from 5 to20 μm. Small and wide angle patterns were recorded sequentially on thesame area by changing the sample-to-detector distance. The diffractionpatterns were recorded with a MARR CCD detector with exposure times of10 and 30 sec for wide and small angle respectively. The detectorfeatures are: 2048×2048 pixels, 64.45 μm×64.451 μm; 16 bit readout. Thethin radial hemisections and transverse sections of fully calcifiedalternate osteons were investigated at various angles with respect tothe incident beam used for verification of results.

Results. A first examination of the experimental images shows thefollowing. The SAXS images of dark lamellae are unchanged in shapewithin scanned areas, evidencing one preferential orientation ofcollagen bundles (FIG. 34). A clear arching of the small-anglemeridional reflection, which corresponds to the third-order collagenperiodicity, is indicative of collagen orientation with respect to theosteon axis. The arching shows no change within and across the scannedareas with intensity preferentially distributed in one direction,indicating a single preferential collagen bundle direction. Moreover,the position of the maximum intensity oriented perpendicularly to thebright lamellar width and the arching of the small-angle meridionalreflection parallel to the bright lamellar width are indicative ofcollagen bundle orientation parallel to the osteon axis (FIG. 35).

The WAXS images of dark lamellae show a consistent preferentialorientation of the 002 reflection parallel to the bright lamellar width,which indicates one preferential orientation of hydroxyapatite patternsalong the osteon axis (FIG. 36). In contrast, the SAXS images of brightlamellae change in shape (intensity and inclination) within scannedareas, evidencing one or two preferential orientations of collagenbundles at ±45° lamellar width direction (FIG. 37). Therefore, brightlamellae contain areas with oblique collagen bundles, at approximately±45° with the osteon axis. The arching of the SAXS meridional reflectionis unclear on bright lamellar specimens. It is therefore indicative of alack of specific collagen orientation. The WAXS images on brightlamellae show a preferential orientation of the 002 reflection at 45°with respect to the lamellar width direction in only some areas (FIG.38). This indicates the local presence of hydroxyapatite orientationoblique to the osteon axis. The lack of consistent preferentialorientation of hydroxyapatite patterns suggests a higher directionaldisorder in comparison with dark lamellae. Note the consistency of theresults for collagen bundles and hydroxyapatite patterns, which supportthe similarity of patterns between collagen and hydroxyapatite.Experimental images are currently being examined for differences inhydroxyapatite density between the two lamellar types.

EXAMPLE 6 Computerized Geometric/Structural Osteon Model

Similar to Example 4, this Example provides a viscoelastic model forosteons based on modeling of experimental data. Osteon specimens areprepared for subjection to dynamic torsional loading and biochemicalanalysis. Such tests yield viscoelastic properties and the relativepercentages of collagen and mucopolysaccharides. The results obtainedforms the basis for a computerized geometric/structural osteon model,that simulates the formation and propagation of fractures observed inthe loaded specimens. The study is described by the following flowchart:

Osteon Specimen Preparation and Selection

Because of the rigorous criteria for the selection of osteon specimens(such as the haversian canal and specimen cylindrical surface beingcoaxial), only about 1 out of 25 specimens isolated may be eligible formechanical testing.

For mechanical and chemical experiments, the bone materialspecifications and specimen preparation techniques are conductedaccording to the specification. Initial and final stages ofcalcification, as evidenced by micro-X-ray (Amprino and Engstrom, 1952),are chosen for the longitudinal and alternate osteon specimens. Thetype, whether longitudinal or alternate, of the specimen can be assessedonly after loading by observation under polarized light a thincross-section cut from the specimen with the dental drill. In fact,because the thickness of the transverse bone section is slightly largerthan the osteon mean diameter, concentric lamellae overlap, therebyreducing or precluding the visibility of dark lamellae, and leaving openthe possibility that an alternate osteon may have a bright appearance.The osteon types can be identified after mechanical testing by cutting a100 μm cross section with the dental drill and examining it underpolarizing light. Therefore, for the mechanical investigation proposed,it is necessary to prepare between 2,700 and 3,000 specimens to obtain120 specimens (30 osteon specimens per osteon type at initial and finalstages of calcification) which will satisfactorily complete theprocedures adopted for loading under torsion at low and high strainrates.

For the biochemical analysis, specimens with the same geometric andstructural characteristics are prepared. The procedure is considerablymore expeditious but requires more specimens. The specimens forbiochemical analysis are isolated from 500 μm thick transverse sectionsusing the technique of Ascenzi A. and Bonucci (1968). In this technique,no lugs are necessary and the type of a 500 μm long osteon specimen canbe easily recognized on transverse sections (FIG. 22). Osteons, whosecanal of 40±3 μm runs perpendicular to the transverse section, aremeasured before isolation in terms of thickness of bright and darklamellae on both sides of the section (see section entitled “ThicknessVariation Within Dark and Bright Lamellae”). The number of bright anddark lamellae is counted at both sides of the section by the imagingsystem. Osteons whose distributions in terms of thickness and number ofbright and dark lamellae match those of mechanically tested osteonspecimens are selected for isolation. The dental drill is used toisolate cylindrical osteons specimens with a 210±3 μm outer diameter bycutting with an off-center tip along a circle. In fact, when therotating axis of the needle is perpendicular to bone section surfaces,i.e. its axis coincides with the osteon axis, the tip of the needle cutsan osteon specimen of cylindrical shape with walls of uniform thickness.After isolation, the length and diameter of each specimen is checked bymeans of an eyepiece micrometer. Further, the Haversian canal is cleanedout by inserting a metal rod slightly (45 μm, approximately) thickerthan the Haversian canal. It is necessary to isolate approximately 100osteon specimens (FIG. 25) per osteon type at initial and final stagesof calcification to have sufficient material volume for each of the twobiochemical analyses.

Because of the different technical requirements of the mechanicaltesting and the biochemical analyses, the test specimens for each comefrom separately prepared groups of osteon specimens. The two groupsconsists of osteon specimens isolated from each region of the samemid-diaphyseal section of the femur in the same proportion. Thestrictness of the criteria for specimen selection and preparationensures homogeneity between the two specimen groups.

Mechanical Testing

Once osteon specimens are prepared as described above, wet osteonspecimens are tested under monotonic torsional loading. Amicro-torsimeter as described above is employed, using directvisualization of the specimen with a stereo microscope during theloading sequence. The osteon specimens are loaded until rupture atconstant low (10⁻²) and high (10 sec⁻¹) strain rates.

For the proposed experiments the torsimeter is modified for attachmentto an MTS 612 servohydraulic testing machine. The fixed stock of thetorsimeter incorporates a reaction torque sensor (5 oz-in capacity). Thedrive shaft is modified to incorporate a bowstring arrangement toconvert the linear actuator displacement to angle of twist. The base ofthe torsimeter is rigidly mounted to centralize the actuator axis withthe bowstring arrangement. Ramp loadings of 10⁻² mm/sec and 10 mm/sec isused to apply torque to the specimens. All the specimens are taken tofailure.

The resulting output is a plot of torque versus angle-of-twist recordedin real time. A direct dependency of the torque's slope versusangle-of-twist curves is observed on the loading rate (see also previousExamples demonstrating a significant creep effect). The experimentaldiagrams are graphs of increasing functions of the angle-of-twist, whichshow a concave downward shape, regardless of the osteon type and thedegree of calcification. From considerations on dynamic behavior andosteon collagen make-up, the diagrams are steeper and less concave forthe high (rather than low) strain rate, independently of osteon type anddegree of calcification; for low strain rate (rather than thequasi-static condition of Ascenzi A. et al., 1994), independently fromthe osteon type at the final degree of calcification; for the osteonspecimens at the final (rather than initial) degree of calcification,independently of osteon type, for any fixed strain rate; and for thelongitudinal (rather than alternate) osteon specimens, independently ofdegree of calcification, for any fixed strain rate.

After the mechanical testing, osteon specimens are examined under anoptical microscope under both regular and fluorescent light (Huja etal., 1999) to assess fracture patterns. At low strain rates, adistortion of the lamellar structures in the areas under compression isobserved (McElhaney and Byars, 1965). No cross-lamellar orcross-osteonal cracks occur because they have not been observed inmacroscopic specimens. That outer lamellae are involved in the fractureprocess is tentatively explained by the decrease in hydroxyapatitepresent in the osteons as one proceeds from the vascular canal to theouter wall (Rho et al., 1999). In both osteon types the fracture agreeswith the observations of Jepsen et al. (1999) that fracture inmacroscopic specimens is ductile and that fracture alters bone's viscousbehavior. In particular, relaxation increases with the increasing extentof fractures. Consequently, viscous effects in osteon specimens canincrease with the increasing extent of fracture. The transition of theshape of our osteon specimens from a circular to a square cross-sectionsuggests a stress concentration effect at the junction of thecylindrical portion with the rectangular lugs situated at the ends.Therefore, fractures can begin at the end of some specimens earlierduring loading than would otherwise be expected.

Further, at a fixed strain rate, longitudinal and alternate osteonsdiffer with respect to fracture patterns. This results from thefollowing combination of general considerations on torque with previousresults on osteons. On one hand, application of torque to a body can beviewed as internal tensile and compressive stresses using a judiciouschoice of material element orientation. The tensile and compressivestresses act approximately at a 45° angle to the longitudinal axis ofthe body. If a material is weaker in a longitudinal orientation than intransverse shearing, the first cracks arise from axial shearing stressesand appear in a longitudinal direction. However, if the material isweaker in tension than in shear, it usually cracks along a spiral courseinclined at a 45° angle to the long axis of the body (Timoshenko andYoung, 1940). On the other hand, Ascenzi A. et al. (1967 and 1972) foundthat at the same degree of calcification longitudinal osteons are weakerin longitudinal than in transverse shearing, while alternate osteons areweaker in tension than in shear. Therefore, in longitudinal osteons thefracture is expected to start somewhat longitudinally, between collagenbundles. Therefore, it can deviate once or twice at the beginning of thefracture and be soon followed by a smooth crack advancing rapidly acrossthe osteon to possibly end in the vascular canal. As torque increases,collagen bundles between cracks are expected to break and crackscoalesce to create one or more predominantly longitudinal cracks.Deviations causing a dentate profile may form due to the viscoelastic,strain rate sensitive collagen and mucopolysaccharides. Such a dentateprofile is therefore be more evident at low (rather than) high strainrate and at the initial stages of calcification when the osteons arericher in mucopolysaccharides. In alternate osteons, cracks spreadobliquely by following the weak interfaces of lamellae. The transverseand oblique collagen bundles may break before the longitudinal ones asthe osteon section enlarges. In this instance, cracks spread throughlamellae less rapidly than in longitudinal osteons, as explained by thecrack propagation control, characteristic of composite materials (Cookand Gordon, 1964). Once the crack breaks through transverse and obliquebundles, it propagates faster straight through the vascular canal. Along crack shows an oblique orientation between upper and lowerextremities. In alternate osteons at initial stages of calcification thecracks may start at a higher value of torque and propagate more slowlythan at the final stages of calcification if the percent ofmucopolysaccharides is higher.

After observation of fracture patterns, the specimens are measured inlength and diameter. After lug removal, the alternate osteons aresectioned and examined to assess the collagen bundle arrangements. Atthat point, longitudinal and alternate osteon specimens only areincluded in the investigation. The dark and bright lamellae oflongitudinal and alternate osteons are counted and measured in thicknessunder polarizing light. Such values are comparable to the ones obtainedby Ardizzoni (2001) by a different method. The knowledge of number andthickness of lamellae within mechanically tested specimens is necessaryto prepare for the biochemical analysis sets of specimens with samedistributions of lamellae.

Biochemical Analysis

A portion of the osteon specimens, prepared and selected as describedabove are subjected to chemical analysis. Spectroscopic techniques andvideo microscopy are used to study intermolecular interactions in phasetransitions. These spectroscopic techniques, coupled withchromatographic separations, are critical in the biochemical analyses.Collagen (as hydroxyproline) and mucopolysaccharides (as hexosamine) isassessed on 150 μg amounts of longitudinal and alternate osteons atinitial and final stages of calcification. While it is difficult toquantify small quantities precisely, the uncertainty does not affect thecomparative conclusions to be made (Herring, 1972).

Osteon specimens are dried to constant weight in a vacuum desiccatorover P₂O₅. Since decalcification of osteons is necessary to quantifyhexosamine and hydroxyproline, acid hydrolysis is used. Residual HCl isremoved before the assays are performed. Hydrolysis products areseparated using refined chromatographic techniques as first described byExley (1957). Hexosamine is determined spectroscopically essentiallyaccording to the procedure of Elson and Morgan, (Exley 1957) as modifiedby Oguchi et al. (1979). The methodology is refined to eliminate thepossibility of interference from amino acids and mineral salts(Pugliarello et al., 1970). Hydroxyproline is determined, essentiallyaccording to the procedure of Serafini-Cessi and Cessi (1965), as laterrefined by Teerlink et al. (1989).

The biochemical analysis has been performed successfully (Pugliarello etal., 1970) on osteon specimens at initial and final stages ofcalcification, without regard to osteon type. Moro et al. (2000) haveemployed on rat bone a technique refinement that can be applied toosteons. Collagen and mucopolysaccharides percentages are significantlylower in longitudinal rather than alternate osteons. In fact, alternateosteons contain more collagen and mucopolysaccharides than longitudinalosteons at equal degrees of calcification, because alternate osteonscontain bright lamellae, which are richer than dark lamellae in collagenand mucopolysaccharides. The mucopolysaccharides percentage decrease, asthe degree of calcification increases, is statistically significant (nonsignificant, respectively) in alternate (longitudinal, respectively)osteons. In fact, mucopolysaccharide percentage needs to decrease inosteons, regardless of the osteon type, for the degree of calcificationto increase. The resulting collagen and mucopolysaccharides percentagesfor longitudinal (alternate, respectively) osteon specimens is lower(higher, respectively) than the values found by Pugliarello et al.,1970) regardless of the osteon type. The means of collagen andmucopolysaccharides percentages is combined with the mean number of darkand bright lamellae in longitudinal and alternate osteons to yield themean percent of collagen and mucopolysaccharides within dark and brightlamellae, at initial and final stages of calcification.

The Two Way Analysis of Variance is applied to the means of the collagenand mucopolysaccharides with osteon type (longitudinal or alternate) anddegree of calcification (initial or final) as factors. If normalitylacks, the 2-way ANOVA is applied to the means of the logarithms.Significance is set at 0.05. The Post hoc Student-Newman-Keuls testidentifies the significant factors. The biochemical analysis shows: astatistically significant higher percent of collagen andmucopolysaccharides in alternate rather than in longitudinal osteonspecimens at equal stages of calcification; a statistically significanthigher percent of collagen and mucopolysaccharides in bright rather thanin dark lamellae at equal stages of calcification; and a statisticallysignificant (not significant, respectively) decreasing amount ofcollagen and mucopolysaccharides as the degree of calcificationincreases in alternate (longitudinal, respectively) osteon types.

Any discrepancies or unexpected results may be of two types. Either anexpected difference is an actual no difference or the difference isreversed. The actual results are examined in relation to the observedmechanical behavior through the computerized model, which is implementedwith the actual chemical percentages. For instance, an actual nodifference result of an elementary component percentage between twospecimen group can mean that such elementary component does notcontribute to a difference in mechanical behavior between such groups.

Analysis of Mechanical Diagrams

The observation of the experimental diagrams assesses elastic modulusand the ultimate strength. For each experimental diagram, theconstitutive equation, which relates stress, strain and their timedependencies, is established in terms of the Ramberg-Osgood equation:

$\theta = {{{Tc}\left( \frac{\mathbb{d}\theta}{\mathbb{d}t} \right)}^{d} + {{aT}^{N}\left( \frac{\mathbb{d}\theta}{\mathbb{d}t} \right)}^{b}}$

where θ and T denote angle-of-twist and torque respectively; a, b, c, d,N denote constant values that depend on the material properties, with a,b, c≧0 and d≦0. This equation has accurately fitted the response set atvarious strain rates of a wide range of tested engineering materials(Hight and Brandeau, 1983). This is because on one hand the geometricshape of the experimental diagrams changes only moderately as the strainrate varies. On the other hand, the Ramberg-Osgood equation is a simplyformulated polynomial whose coefficients are functions of the strainrate, which describes the relatively simple geometric shape of theexperimental diagrams, i.e. an increasing function graph, that passesthrough the origin and is concave down.

The determination of the constants a, b, c, d and N follows theprocedure used by Hight and Brandeau (1983) for macroscopic compact bonespecimens. The Ramberg-Osgood equation suffices to produce a good fit(measured by an r² of 0.98-0.99) because the viscoelastic behavior of asingle isolated osteon specimen is less complex than that of macroscopicspecimens. In cases where the Ramberg-Osgood equation does not produce agood fit, increasingly more complex differential equations can beemployed, starting with more complex polynomials and rational functionsof T whose coefficients depend on the strain rate. Linear-viscosity isobtained (Frasca et al., 1977) at least for physiological strains. Ifthe Ramberg-Osgood equation provides a good fit and linear-viscosity ispresent, the coefficients a, b, c, N equal 0, 1, 0, 0, respectively. Theapproximating function of each experimental diagram serves to computethe viscoelastic modulus as the derivative of the diagram approximatingfunction at zero strain and the energy absorption capacity as the areaunder the approximating curve.

The 2-way ANOVA is applied to the means of the elastic and viscoelasticmoduli, ultimate strength and energy absorption with osteon type(longitudinal or alternate) and degree of calcification (initial orfinal) as factors for each strain rate. If normality lacks, the 2-wayANOVA is applied to the means of the logarithms. Significance is set at0.05. The Post hoc Student-Newman-Keuls test identifies the significantfactors. Significant differences are obtained in elastic andviscoelastic moduli, ultimate strength, and energy absorption capacitywith regard to the following: higher values at the final stages ratherthan at the initial stages of calcification, regardless of osteon typeand the strain rates; higher values for longitudinal rather thanalternate osteon specimens, regardless of degree of calcification andthe strain rate; higher values for longitudinal rather than alternateosteon specimens, regardless of degree of calcification and the strainrate; smaller increase in value for longitudinal rather than alternateosteon specimens, as the strain rate increases, regardless of degree ofcalcification; and increase in values with increasing strain rate,regardless of osteon type and degree of calcification.

Geometric/Structural Osteon Specimen Model

The osteon specimen model fits the mechanical behavior of the osteonspecimen to that of its ultrastructural components. Therefore, the modelis based on the mechanical diagrams' approximating functions, angle-oftwist as function of torque, and the relative percentages of theultrastructural constituents, as obtained from the biochemical analysis.This osteon specimen computerized model allows for the concomitantincorporation of elementary component percentages and orientations, andtheir role during visco-elastic and plastic phases. The model allows forsimulation of the fracture propagation observed in the mechanicallytested specimens. In particular, the model sheds light as to theprocesses within the osteon specimen of micro-cracking, de-bonding, poregrowth and components' breakage, which result in fracture patterns.

The model's crucial parameters are assessed from the specimens employedfor mechanical and chemical analysis. Other important parameters, i.e.collagen bundle and hydroxyapatite pattern directions and brightlamellar prestress, have been investigated in large numbers (on theorder of a few hundreds so as to cover the biological variability ofosteon structures). Additional parameters, i.e. dimensions anddistribution of canaliculae and lacunae, viscoelastic properties ofcollagen and mucopolysaccharides, collagen bundle diameter, and elasticproperties and density of hydroxyapatite have not been assessed inspecimens and are addressed by treating them as parameters which vary,within ranges consistent with lamellar structures, around availablevalues from the literature. The ranging of such additional parametersaccordingly serves to model different scenarios.

Material science implies that some of these parameters of a fiberreinforced laminated model have a role (perhaps minor) in the modeledcomparative behavior of the two osteon types. For instance, thepresence, relative distribution and density of canaliculae and lacunaeaffects the fracture simulation patterns, rather than the particulardimension or position of a canalicula or lacuna. Accordingly, theparametrized model of the invention allows for different values of theparameters, either exclusion of canaliculae and lacunae altogether orvariation in porosity distribution. Since there is no evidence ofdifference in the literature between osteon types with respect toviscoelastic properties of collagen and mucopolysaccharides, and elasticproperties of hydroxyapatite, their parametric values are kept constantacross the two osteon model. Such parametric values are less relevantbecause of the comparative nature of the model. The starting parametricvalues of the hydroxyapatite density follow any relevant findings from acomplete examination of X-diffraction results and indications from theliterature. The initial parameter value for collagen bundle diametermeasurement may be obtained from the literature and is kept constantacross the osteon types.

The geometric model of the osteon specimens before mechanical testingconsists of a hollow cylinder with coaxial lateral surfaces. Itsinternal diameter, external diameter, and height equal 40 μm, 210 μm,and 500 μm, respectively. Each such hollow cylinder presents pores, asshown in FIG. 26. Pores in the model will include haversian canal,canaliculae and lacunae, and the starting parametric value of the totalosteon volume (Piekarski, 1970) is set equal to 20%.

The material model of each of the longitudinal and alternate osteonspecimens before mechanical testing consists of a laminate whose length,width, and height correspond to the cylinder circumference, thickness,and height, respectively (FIG. 27). The layers are unidirectionalfiber-reinforced laminae. The matrix represents a mixture ofhydroxyapatite and mucopolysaccharides. The fibers represent collagen.The three materials that make up matrix and fibers are each treated ashomogeneous and isotropic. Both matrix and fibers are considered asviscoelastoplastic. The fibers are assumed to be circular incross-section and randomly distributed in the transverse plane. Thelongitudinal and alternate osteon models consist of laminae whose numberand thickness will equal the mean of the lamellar number and thicknessof the tested specimens. The lamina with fiber inclination γ is namedγ-lamina. The values of γ will follow the results of the studiesdescribed above, so as to include and exclude, respectively, obliquefibers in the laminae modeling dark lamellae. The initial parametricvalue for the fiber (modeling collagen) diameter is set at 800 Å. Therelative percentages of collagen and mucopolysaccharides at initial andfinal stages of calcification, is quantified by the biochemicalanalysis. The initial parametric percentage of hydroxyapatite at thehighest degree of calcification is set to 40% of the lamina volumewithout pores (Bonfield and Li, 1967). The initial parametrichydroxyapatite volume is set 10% higher in the dark lamellar than in thebright lamellar model (Marotti et al., 1994). The initial parametricvalues that describe the elastic properties of the hydroxyapatite equalthe findings of Katz and Ukraincik (1971). The initial parametric valuesthat describe the viscoelastic properties of the fibers, which modelcollagen, equal the findings of Currey (1959) and Haut (1983). Noinformation is available in the literature in reference to theviscoelastic properties of mucopolysaccharides. Viscoelastic parametricvalues are deduced from similar substances. Little information isavailable regarding the fluids within the pores incorporated in themicrostructure. Initially, the model, by assignments of minimal materialproperty values, disregards the structural effects of the fluid withinthese pores. This form of the model is then be exercised parametricallyto include fluid within the pores with various bulk moduli.

The starting number of elements for the Finite Element Analysis is inthe order of 618,137. This was computed by the use of brick elements ofdimensions averaging 3 μm to have elements comparable to lacunae involume. The element mesh is refined to achieve convergence of thesolution. For any given value of the experimental torque, a computerprogram based on Montecarlo simulation is written to compute stress,strain, and phase deformation distributions. Since strain is chosen asthe criterion for osteon fracture (Piekarski, 1970), the strain withineach element is compared to the yield strain provided by the literature.If cracks appear to initiate at the matrix-fiber interface, anappropriate failure criterion (e.g., Von Mises, Tresca) is included. Theincreased probability of fracture in the neighborhood of an alreadyfractured element is considered using the concept of stress enhancementfactors. The model is verified by checking that the osteon modelsimulates the osteon quasi-static behavior (Ascenzi A. et al., 1994) andthe specimens' dynamic fracture patterns.

The model can evidence the following experimental observations onfracture propagation by various authors. Fractures start at a weakerpoint of the bone structure (Carter et al., 1981). 3 to 4 small cracksform in the organic phase which yields and/or buckles and inhydroxyapatite, which pulls and makes the cracks spread at the weakinterfaces between two outer lamellae (e.g. Piekarski, 1970; Simkin andRobin, 1974). Lacunae' size precludes them from acting as fractureinitiators until or unless plastic deformation has created cracks at thetip and thereby extended them to the critical length for spontaneousfracture. At the front of the propagating crack, the large strains,which may be accommodated by the organic phase, contribute to thedissipation of energy. Microcracks form ahead of the advancing fractureline. The areas containing transverse and oblique collagen bundles canshow slow propagation of cracks to allow for the area to absorb a largeamount of energy. Slow propagation is essentially a pull-out typemechanism, that is, hydroxyapatite crystallites would be pulled out ofthe collagen by shear failure at the fiber-matrix interface. The rapidpropagation of cracks in areas containing approximately verticalcollagen bundles would allow for very low energy absorption. This shouldbe compatible with larger areas under the experimental graphs ofalternate osteons. The propagating crack avoids discontinuities(Piekarsky, 1970), hence increasing its length. Crack propagation isarrested by the presence of canaliculae and lacunae. In the case wherethe crack enters a discontinuity, its front is blunted, hence reducingthe stress concentration factor and slowing crack propagation. When acrack is forced to enter the vascular canal, the radius at the tip ofthe crack becomes larger. Lacunae are more likely to act as stressconcentrators than canaliculae because of their ellipsoidalcross-section. Since porosity acts as a crack arrester, porosity maycontribute to increase bone's robustness (Currey, 1962) rather thanincrease its tendency for brittle fracture. Hydroxyapatite crystallitescan be pulled out from collagen around canaliculae.

The proposed model simplifies the osteon structure; in particular:exclusion of partially calcified collagen bundles (Ascenzi A. et al.,1965) because they are not yet well understood in terms of structure;assumption of no difference in collagen mechanical properties betweendark and bright lamellae because no evidence is currently available andmaterial science points to differences in fiber bundle directions asprimarily responsible for structural differences; assumption of nodifference in dark and bright lamellae, separately, between longitudinaland alternate osteons because there is no evidence of difference and nolamellar isolation technique is available for longitudinal osteons; andassumption of constant lamellar thickness within any given osteonlamellae, for simplicity.

EXPERIMENTAL DESIGN OF EXAMPLES 7-14

The experiments of Examples 7-14 described herein were designed toresolve the conflicting views on the ultrastructure of bone, inparticular lamellae, and to investigate whether the mechanical responseof single lamellae to tensional loading can depend on the lamella'smorphology with respect to as collagen bundle orientation and densitydistribution.

For this purpose, the following study outline was prepared:

(1) Fully calcified isolated lamellae, distinguished according to theirappearance as either bright or dark under transverse polarized light,will be investigated by confocal microscopy. Domains, i.e., regions ofessentially unidirectional collagen bundles, are identified.Measurements include dimensions of domains and of spaces betweendomains, and collagen bundle direction and density within domains.

(2) Specimens of lamellae of the same specification of those undergoingconfocal examination are tested in distinct orthogonal directions todetermine differences in structural response as a function of loadingdirection. This is accomplished using precision controlled quasi-statictensional testing.

These measurements increase the understanding of the constituent make-upof bone at the single lamella level and provide a prospective on theinfluence that this make-up has on the lamellar mechanical behavior. Anumerical model of the fully calcified osteonic lamella that accuratelyreflects the characteristics of geometry, distribution and orientationof its components, yields a realistic simulation of the quasi-statictensional behavior observed experimentally.

(3) A finite element model is produced reflecting the geometry,distribution and orientation of the identified collagen bundles, asestablished in (1). In addition, this model incorporates porosity toproperly reflect the microscopic structure of the single lamella.

(4) The model created is parametrically exercised within the limits ofknown property variation in literature and set to conform to theexperimental results to allow for biological variations and to studybiological effects. This includes variation of the experimentallymeasured elastic constants as well as manipulation of the loaddistribution and boundary conditions. Finally, the effects of theseparametric manipulations on fracture and failure characteristics isinvestigated.

The above studies enable realistic representation of the geometry andthe structure at the level of a single lamella. The representation ofindividual units can also be integrated to increae the accuracy of themechanical behavior of osteons and osteon groups.

Overview of Methods and Results

To establish the microstructural differences in terms of lamellarelementary components and the impact of such differences on lamellarmechanical behavior, appropriate methodology is needed. In theexperiments described herein, multiple methods of investigation wereemployed to shed light from different perspectives on the morphologicalarrangement of lamellar specimens; 1) circularly polarized lightmicroscopy; 2) confocal microscopy; and 3) X ray diffraction methods.Confocal microscopy and small-angle X ray diffraction were employed onlamellar specimens. Other useful methodologies include combinations ofconfocal microscopy examinations and mechanical test procedures throughmathematical analysis and computer modeling, to examine theultrastructural variables, which regulate mechanical response to appliedtension.

The studies can utilize, for example, two external lamellae of osteonsat final stages of calcification obtained from donors in the 27-49 yearold age. Osteons that exhibit the final stages of calcification, asevidenced by micro-X-ray (Amprino and Engström, 1952), are chosen forthe alternate osteon specimens. Polarized light microscopy is used toconfirm that the selected osteons are of the alternate osteon type.According to Rho et al. (1999), there is no evidence that the lamellarelastic modulus of external lamellae in fully calcified osteons dependson either osteon size or osteon lamellar number. Therefore, the degreeof calcification of the external lamellae is not an important variablewithin and across the lamellar specimen groups.

To conduct experiments on osteonic lamellae, the availability ofisolated osteonic specimens from the mid-diaphysis of the human femur ofappropriate type and sufficient quantity is especially advantageous. Themid-diaphysis of long bones has been shown to contain osteons that aremost consistent from a structural and morphological perspective relativeto those obtained from other cortical bone locations. Osteonic lamellaevary in thickness (2-11 μm) and length (up to a few centimeters). Theycomprise the generally coaxial cylindrical layers of osteons, whichmeasure 200-250 μm in diameter. The number of lamellar layers that makeup an osteon range from 4-20. For a meaningful systematic study, allspecimens need to have the same dimensions and structuralcharacteristics while maintaining the characteristics of the grossentity from which they are selected as specimens. Thus, the variablescan be dealt with realistically as follows:

Collagen bundle make-up: Microscopic examination of compact bonesections under polarized light reveals that all secondary osteonsobtained from the mid diaphysis are composed of coaxial lamellae thatappear dark and lamellae that appear bright. Examination of thousands ofbone sections has revealed the following consistent observations: 1)osteons made up exclusively with bright lamellae are uncommon; osteonsthat are mostly dark in appearance show a thin layer of bright lamellaearound the harvesian canal; and 3) among all the different combinationsof dark and bright lamellae in osteons, osteons made up of essentiallydark lamellae (so-called longitudinal osteons) and osteons made up ofalternatively dark and bright lamellae (so-called alternate osteons)represent two ends of a spectrum, biologically and mechanically. Sincethe lamellar type classification, according to whether they are dark orbright, are numerous, and they appear to be independent from the osteontype to which it belongs, investigations can, at least initially, beconcerned with alternate osteons only, i.e., those that revealalternating lamellae represented by dark and light fields.

Degree of calcification: Examination of micro-X-rays of compact bonesections reveals a spectrum of intensities, ranging from dark gray towhite. The X-ray intensity indicates the degree of calcification of anosteon as a whole from initial to final (Amprino and Engström, 1952).Osteons at the final stage of calcification constitute the majority,close to perhaps 90% of adult compact bone. The level of hydroxyapatitein osteons at the final stages of calcification is usually considered toremain constant within the adult age group (27 to 49 years). Severalstudies have indicated that the degree of calcification decreases withinosteons radially from the haversian canal to the cement line (e.g.,Amprino and Engstrom, 1952; Skedros, et al. 1996).

Lamellar thickness: The distribution of lamellar thickness varies amongand within individual osteons. Lamellar thickness varies from lamella tolamella from the cement line to the haversian canal (Rho et al., 1999;Ardizzoni, 2001) and within the lamella along the osteon length (AscenziA. et al., 1982). Lamellar thickness measurements remain comparable tothe values obtained on thinner sections by electron microscopy (AscenziA. et al., 1982) to an extent of approximately 100 μm of its length.Because of these expected normal variations, length and width ofspecimens need to be uniform within and across the specimen groups. Theselected dimensions need to be large enough so as to provide for a goodrepresentation of the underlying lamellar structure.

To observe isolated lamellae, embedded in and isolated from thealternate osteon (the osteon that appears alternatively dark and brightin cross-section under the polarizing microscope), a novel isolationtechnique was applied. The novel isolation technique enabled examinationof the two dissected lamellar types through and across their flattenedcylindrical surface by circularly polarized microscopy, confocalmicroscopy and X-ray diffraction. As described in the Examples, it wasevidenced that:

-   -   embedded lamellar thickness varies along the lamellar length,        therefore, thickness should be measured along both lamellar        specimen edges;    -   embedded lamellar thickness varies between dry and wet        condition, hence all measurements should be taken on wet        lamellar specimens;    -   polarized light microscopy indicates the presence of a        relatively continuous orientation pattern of oblique collagen        bundles within the bright lamella and of a more discrete pattern        of orientations within the dark lamella;    -   confocal microscopy indicates morphological differences between        the two lamellar types and indicates the appropriateness of a        more detailed confocal microscopy study with finer scanning;    -   X ray diffraction supports an inference of the presence of        longitudinal collagen bundles in dark lamellae and of oblique        collagen bundles in bright lamellae.

Mechanical testing of specimens in tension can further support thefindings listed in the last three points, and correlate to anystructural differences observed in the confocal study.

Lamellar Specimen—Selection and Preparation

Lamellar specimens are prepared for tension loading and confocalmicroscopy analysis. Such tests yield mechanical properties and collagenorientation patterns, respectively. The results obtained will form thebasis for a computerized geometric/structural osteon model, that willsimulate the formation and propagation of fractures observed in theloaded specimens.

The ability to complete lamellar specimen preparation and selection is aprerequisite to conduct the described experiments. Because of thedifficulty of the specimen preparation, it may be expected that onlyabout 1 out of 3 specimens isolated are eligible for mechanical testing.

The first prerequisite to provide a uniform sample selection is tominimize variability among the specimens that will be included forevaluation. Initially transverse sections of the whole bone, 30 mm inlength, is obtained from the mid-diaphysis of adult femurs (27-49 yearsof age) that are free of evident skeletal pathologies. These initialspecimens are prepared as described above using a water cooled rotatingsaw blade. The experimental methodology is directed at narrowing thepossible alternatives that describe the underlying structure andcomposition of materials that form the individual osteon lamellae.

The following are brief descriptions of the procedures that areemployed. Polarized light microscopy studies have been limited toobservations on the transverse sections of intact osteons and throughthe thickness of individual isolated lamellae. Lamellae that appearbright under these conditions indicate a preferred structural alignmentof collagen bundles and hydroxyapatite in a plane perpendicular to theaxis of polarization. If the alternating bright and dark fields areindicative of structural arrangement then the appearance of the layersshould be reversed when the same sample is viewed from thecircumferential direction in the direction of polarization. Peliminaryconfocal microscopy studies is optionally performed at sufficientresolution to distinguish explicit differences in structure on the orderof the size of individual collagen bundles. Scanning at 0.25 mmincrements enables distinguishing distinct collagen bundle arrangementsthat are present in the two types of lamellae. Identification throughimproved detail will enable reconstruction of the three dimensionalnature of these constituents' geometry that would indicate a preferencefor resistance to directional loading.

Mechanical testing of individual lamellae have only been performed on asingle type in tension and only in a single direction (equivalent to itsoriginal circumferential alignment). The loading of a single lamellatype in both the circumferential and longitudinal directions willprovide direct mechanical evidence of differences in mechanicalbehavior. It will also elucidate differences in the mechanical behaviorbetween lamellar types. Finally, our detailed analytical modeling willenable us to parametrically exercise possible existing variations inarchitecture and elemental composition of the individual lamellararrangements and the influence that these arrangements have on themechanical response to applied loadings.

The combination of these procedures will enable us to confirmdifferences in structure and composition related to mechanical behavior.It will further enable us to generate refined representations of theorganizational structure of individual lamellar layers and resolve manyof the current discrepancies.

The lamellar specimens are isolated in the longitudinal orcircumferential (anatomical) dimensions; the lamellar thickness will bealways measured in the original radial direction; specimen length,width, and thickness will refer to largest, intermediate, and smallestspecimen dimension. Width and length are measured at several locationsfor each sample using a calibrated image processing system (Image-1).The results are averaged for each dimension. The thickness of eachspecimen is measured using commercially available distance transducers(digital displacement inductive3 sensor, Keyence model EX 305). Severalmeasurements are obtained of the thickness of the glass-specimencomposite along the length of the specimen. Separate measurements of theglass thickness are also obtained. Subtraction of the net glassthickness from the composite measurements yields specimen thickness.

Confocal Microscopy—Samples

For confocal examination, 50 dark and 50 bright lamellar specimens aredissected from 70±3 μm transverse sections. They consist ofapproximately half of the lamellar circumferential length to allow forexamination of the structure along the longest specimen that can beisolated. For mechanical loading, 60 dark and 60 bright lamellarspecimens are prepared as follows. Dark and bright lamellar specimensare chosen on 115±3 μm and 70±3 μm transverse femoral mid-shaft sectionsin terms of their mean thickness along the two edges. They are measuredwet inside the alternate osteon along the circumference and at the ends,as described in the studies above. Dark and bright lamellar specimensare chosen so that the wet mean thickness along each edge equals 3±1 μmand 4±1 μm, respectively. Dark and bright lamellar specimens aredissected wet as described in section 3.1 along an arc length of 70±3 μmon 115±3 μm thick transverse sections and along an arc length of 115±3μm on 70±3 μm thick transverse sections.

The 60 specimens of each lamellar type are chosen so that the lengthdirection of 30 of the specimens is longitudinal and the lengthdirection of 30 specimens is transverse with respect to the osteon axis.This is to determine the mechanical response of each lamellar type withrespect to loading in mutually orthogonal directions that correspond totheir original longitudinal and transverse anatomical orientations.Loading is always directed along the long dimension of the specimen.

While preliminary confocal microscope observations indicate differenceswithin lamellae of the same type with respect to pattering (e.g. domainsize, number, and collagen orientation within domains), the mechanicalproperties within each lamellar group type are independent of the domainfeature variation. This is because the mechanical response depends onthe overall contributions of overlapping domains throughout thethickness dimension. This ensures enough structure homogeneity acrossthe samples of the two lamellar type group separately. The specimendimensions for mechanical testing set at 115 μm×70 μm are dictated bytechnical limitations. While the ratio of the two dimensions is smallerthan the ratio recommended by engineering textbooks to avoid shear andboundary effects during tensional loading along the largest dimension,the comparative nature of the experiments make the lamellar specimenratio acceptable. The shear and boundary effects are addressed in thenumerical lamellar model. The specimens prepared for confocal microscopeexamination and for mechanical testing will consist of lamellarspecimens isolated from each region of the same mid diaphyseal sectionof the femur in the same proportion to ensure homogeneity across the twogroups.

All lamellar specimens are inspected with an optical microscope formicro-cracks after preparation and prior to confocal microscope analysisand mechanical loading so as to ensure integrity of the samples. Sincethe specimen micro dimensions make it easy to lose the specimen and thefragility of the lamellar specimens makes them prone to fracture, forthe investigation proposed, it is necessary to prepare between 1,000 and2,000 specimens to obtain a yield of about 220 specimens (110 lamellarspecimens per lamellar type at final stages of calcification) which willsatisfactorily complete the procedures adopted for examination underconfocal microscopy and quasi static tensional loading.

Confocal Microscopy—Analysis

Dark and bright lamellar specimens are scanned every 0.25 μm throughtheir thickness. Scanning covers the full extent of the length and widthof the specimen. Collagen bundle orientation and density are measured bymeans of an imaging system. A weighted average of the structure isdetermined according to a range of orientations.

Also performed are two dimensional spatial fast Fourier transforms (2DSFFT) of the images to discretize the information into an amplitudeversus frequency spectrum. The areas of domains, i.e., regions ofessentially unidirectional collagen bundles, and areas between domainsare measured. Collagen bundle density is computed from the distances ofadjacent collagen bundles within any given domain. Means and standarddeviations are computed at each scanning level across the lamellargroups, separately. Each domain is associated with a mean collagenbundle orientation by averaging the orientation of its individualcollagen bundles. A finer discretization is considered if warranted.Domains are classified in terms of frequency bins of 15° rangeorientation intervals and in terms of mean collagen density.

Two-way Analysis of Variance (ANOVA) is applied to the means of collagenbundle orientation and density, and domain area with depth level andlamellar type (dark or bright) as factors. If normality lacks, the 2-wayANOVA is applied to the means of the logarithms. Significance is set at0.05. The Post hoc Student Newman Keuls test identifies those factorsthat are significant.

Significant differences in collagen bundle orientation pattern anddensity, domain area and area between domain in the thickness directionbetween dark and bright lamellar specimens are found. Specifically, darklamellar specimens are shown to consist almost exclusively of collagenbundles transverse to the osteon axis perhaps with additionally thinlayers of oblique collagen bundles at the edges. The structure of thedark lamella along its circumferential surface should be analogous tothe one of the bright lamellar specimens because bright lamellae areadjacent to dark lamellae in the alternate osteons. The bright lamellarspecimens will show collagen bundles transverse to the osteon axistowards the middle thickness with oblique collagen bundles graduallyincreasing in angle towards the outer lateral boundaries. Collagenbundle density and area between domains values should be lower andhigher, respectively, in dark rather than bright lamellar specimensbecause various authors (e.g., Marotti, 1990) have hypothesized thatdark (bright, respectively) lamellae are collagen poor (rich,respectively). In particular, lamellae of both types consist of variousdomains, as indicated by Boyde (1972) by means of scanning electronmicroscopy.

A Leica TCS SP inverted confocal microscope with an integralcomputerized imaging system is used.

Mechanical Testing

Dark and bright lamellar specimens are tested under monotonicquasi-static loading as follows. A micro-extensometer is used (AscenziA. et al., 1982) with direct visualization of the specimen with a stereomicroscope during the specimen set up and loading sequence. The lamellarspecimen is secured to the extensimeter. In particular, each end of eachflattened lamellar sample is attached to the end of a support. Each 80μm wide support is cut from a copper grid typically used in electronmicroscopy (FIG. 49). A very strong and quickly drying glue (Kemi CyakCF type) is used to attach the end of each lamella to each support. Careis taken so that the glue covers a minimal portion of the lamella.Glue-free lamellar length is measured under wet conditions to ensuresame length across the dark and bright lamellar specimen groups. Thefree ends of each support are then glued to the micro extensometer. Thealignment is checked by means of the stereomicroscope. The specimens areloaded wet to rupture. For all the specimens, the mechanical loadingdirection is parallel to the length direction.

The resulting output is a plot of load versus deflection recorded inreal time. The experimental diagrams are graphs of increasing functionsof the deflection, which show a concave downward shape, regardless ofthe lamellar type. If confocal microscopy shows that there is nodominant collagen bundle alignment with the loading direction ofspecimens within a lamellar type, composite theory indicates that theplots relative to the longitudinal and transverse specimens of thatlamellar type are expected to show no difference. If instead a dominantcollagen alignment with the loading direction is present, then the plotsrelative to the longitudinal and transverse orientations of thespecimens of that lamellar type are expected to show a difference. Themechanical testing results are expected to correlate with themorphological confocal microscope observations for the two lamellartypes.

After the mechanical testing, lamellar specimens are examined under anoptical microscope under both regular and fluorescent light (Huja etal., 1999) to assess fracture patterns. We expect microcracks to bepresent in the vicinity of pores (canaliculae and lacunae) and at pointsof material weakness and at the interface between elementary components.Evidence of significant fracture density is reflected in reducedstiffness and ultimate strength of the load vs deflection diagrams.Because of the fibered nature of the tissue, cracks are expected toprogress in distinct steps with significant branching and directionalchanges. We will classify the complete rupturing fracture somewhatperpendicular to the loading direction in terms of number of directionalchanges and total length. The mechanical behavior in tension is expectedto be in line with the results obtained by Ascenzi A. et al., (1998) forbright lamellae isolated by a technique different from the one usedhere.

After mechanical testing, the wet specimens are measured in width andthickness by the same methods used before mechanical testing because thespecimens length at rupture is computed from the length beforemechanical testing and the deflection at rupture is provided by the loadvs. deflection plots. The specimen measurements before and aftermechanical testing are necessary to prepare the computerized lamellarmodel.

Analysis of Mechanical Diagrams

Analysis of the experimental diagrams enables determination of themechanical properties of the individual lamellar types. If anexperimental diagram shows a decreasing portion of the graph followingthe expected increasing portion, it means that the specimen hasexperienced large strains. In that case, the specimen sectionperpendicular to the loading access cannot be considered as constant.Therefore, stress vs. strain diagrams cannot be obtained from the loaddisplacement diagrams, and classical theory cannot be employed. In thatcase, the mechanical properties are obtained from the load displacementdiagrams. If instead, an experimental diagram is always increasing, thenthe force vs. displacement diagrams can be converted to stress vs.strain diagrams.

In either case, the constitutive equation which represents thecharacteristic behavior of the curve is established in terms of apolynomial equation of degree three or four. Degree four is expected tosuffice because such degree has accurately fitted the response of fullycalcified osteons to loading in tension (Ascenzi A. et al., 1997), andthe structure of the individual lamellae is simper than that of theosteon. If degree four does not produce a good fit, a higher degree isemployed. The approximating function of each experimental diagram servesto compute the material stiffness as the derivative of the diagram, theenergy absorption capacity as the area under the approximating curve.Ultimate strength (the limit of the approximating functions) is directlymeasured.

The 2-way ANOVA is applied to the means of the elastic modulus, ultimatestrength and energy absorption with lamellar type (dark or bright) asfactor. If normality lacks, the 2-way ANOVA is applied to the means ofthe logarithms. Significance is set at 0.05. The Post hocStudent-Newman-Keuls test will identify the significance of the factors.We expect significant differences in elastic modulus, ultimate strength,and energy absorption capacity for dark rather than bright lamellarspecimens because of more collagen bundles present in dark rather thanbright lamellae.

Geometric/Structural Lamellar Specimen Model

The lamellar specimen model fits the mechanical behavior of the lamellarspecimen to that of its ultrastructural components. Therefore, the modelis based on the approximating functions of the mechanical diagrams,deflection as function of load, and the confocal microscopeobservations. For the first time, this lamellar specimen computerizedmodel allows for the incorporation of domains and their collagen bundleorientations. The model allows for simulation of the fracturepropagation observed in the mechanically tested specimens. Inparticular, the model sheds light as to the processes within thelamellar specimen of micro-cracking, de-bonding, pore growth andelemental component breakage, which contribute to the observed fracturepatterns.

The model's crucial parameters are assessed from the specimens employedfor mechanical and confocal microscope analyses. Other importantparameters, i.e., hydroxyapatite pattern directions and bright lamellarprestress (Ascenzi, M. G., 1999), have been investigated in largenumbers (on the order of a few hundreds so as to cover the biologicalvariability of osteon structures) of specimens as described in thepreliminary studies and previously in the Ascenzi laboratory. Additionalparameters, i.e., dimensions and distribution of canaliculae andlacunae, elastic properties of collagen and mucopolysaccharides,collagen bundle diameter, and elastic properties and density ofhydroxyapatite have not been assessed in specimens and are addressed bytreating them as parameters which vary, within ranges consistent withlamellar structures, around available values from the literature. Theranging of such additional parameters will accordingly serve to exploreand discuss possible scenarios.

Material science implies that some of this fiber reinforced laminatemodel's inherent structure will turn out to have a minor role in themodeled comparative behavior of the two lamellar types. For instance,the presence, relative distribution and density of canaliculae andlacunae will affect the fracture simulation patterns, rather than theparticular dimension or position of a canalicula or lacuna. Confirmationor refusal will derive from the parametrized model that allows, fordifferent values of the parameters, either exclusion of canaliculae andlacunae altogether or variation in porosity distribution. Since there isno evidence of difference in the literature between osteon types withrespect to elastic properties of collagen and mucopolysaccharides, andelastic properties of hydroxyapatite, their parametric values are keptconstant across the two lamellar model. Such parametric values are lessrelevant because of the comparative nature of the model. The startingparametric values of the hydroxyapatite density will follow any relevantfindings from our complete examination of our X diffraction results andindications from the literature. The initial parameter value forcollagen bundle diameter measurement comes from the literature and iskept constant across the osteon types. If confocal microscope imagesyield new information on collagen bundle diameter, such information isincorporated in the model.

The geometric representation of the lamellar specimen consists of threestages. At the first stage, a cylindrical shell represents a wet lamellabefore the dissection of the specimen. The dimensions of the cylindricalshell correspond to the mean measurements of the embedded wet lamella.At the second stage, a parallelepiped represents the wet flattenedlamellar specimen before mechanical testing. The length, width andthickness of the parallelepiped will equal the mean length, width andthickness measured on the wet flattened lamella. At the third stage, aparallelepiped broken into two portions will represent the wet rupturedlamellar specimen after mechanical testing. The dimensions of theparallelepiped will equal the mean measurements of the wet flattenedlamella.

Pores in the model will consist of a distribution of canaliculae andlacunae distributed as described in the literature (Piekarski, 1970).Each of the dark and bright lamellar specimens consists of a two phasefiber reinforced material where the fiber represents the collagenbundles and the matrix represents a mixture of hydroxyapatite andmucopolysaccharides. In this first model, the fiber orientation anddensity is dictated by the results of the lamellar specimen examinationby confocal microscopy. The three materials that make up matrix andfibers are individually treated as homogeneous and isotropic. Theelastic modulus for matrix and fibers are not available, and the valuesused are the best approximation available from the literature. Thefibers are assumed to be circular in cross section and are distributedaccording to the domains observed in our confocal microscopeexamination. The domain with fiber inclination is named g domain (FIG.50). The initial parametric value for the fiber diameter is set at 800Å. The initial parametric percentage of mucopolysaccharides is set equalto 1% (Herring, 1972). For hydroxyapatite the starting percentage is setto 40% of the lamina volume without pores (Bonfield and Li, 1967). Theinitial parametric hydroxyapatite volume is set 10% higher in the darklamellar than in the bright lamellar model (Marotti et al., 1994). Theinitial parametric values that describe the elastic properties of thehydroxyapatite will equal the findings of Katz and Ukraincik (1971). Theinitial parametric values that describe the elastic properties of thefibers, which model collagen, will equal the findings of Currey (1959)and Haut (1983). Since the elastic modulus of collagen, hydroxyapatiteand mucopolysaccharides are not available, the starting values for thematrix and fiber properties are represented by the best approximationcurrently available (e.g., elastic modulus of collage from tendons). Thestarting values are selected from the best available approximations inthe literature. Because fluid is not likely to be retained in thecanaliculae and lacunae in our specimens, fluid will not be consideredto have an influencing role on the structural behavior observed in ourmechanical tests. Therefore, this model will not include stiffeningaffects during specimen loading. This is different from a lamellar modelwhich considers the lamella as part of the intact osteon.

The starting number of elements for the Finite Element Model (FEM) is ofthe order of 500,000. This number was computed by the use of brickelements of dimensions averaging 1 μm, the diameter of a largecanalicula. The element mesh is refined to achieve convergence of thesolution. The FEM will compute the lamellar stress and straindistribution due to the deformation of the cylindrical shell portioninto the parallelepiped before mechanical loading. Such distributionswill take into account the prestress present in dark embedded lamellae(Ascenzi A. and Benvenuti, 1977; Ascenzi M. G., 1999). Then, for anygiven value of the experimental tensional load, a computer program basedon Montecarlo simulation is written to compute stress, strain, and phasedeformation distributions. Since strain is chosen as the criterion forosteon fracture (Piekarski, 1970), the strain within each element iscompared to the yield strain provided by the literature. If cracksappear to initiate at the matrix fiber interface, an appropriate failurecriterion (e.g., Von Mises, Tresca) is included. The increasedprobability of fracture in the neighborhood of an already fracturedelement is considered using the concept of stress enhancement factors.

Fractures that develop within our model are expected to start at aweaker point of the bone structure (Carter et al., 1981). Cracks areexpected to initiate in the organic phase, which yields, and athydroxyapatite crystallite boundaries, which pulls away from the organicmatrix and causes the cracks to advance through the weak interfacesbetween domains where collagen bundles change orientation (Boyde, 1972).Cracks may also initiate at individual lacunae due to a local stressconcentration in the surrounding tissue since these are relatively largewith respect to the smallest dimension of the lamellar specimens. At thefront of the propagating crack, the large strains, which may beaccommodated by the organic phase, are expected to contribute to thedissipation of energy (Simkin and Robin, 1974; Minns, et al, 1973).Numerous microcracks are expected to form throughout the specimen. Theregions containing transverse and oblique collagen bundles would tend toimpede crack advancement and will slow propagation of cracks withgreater energy dissipation. Slow propagation is essentially a pull outtype mechanism, that is, hydroxyapatite crystallites would be pulled outof the collagen by shear failure at the fiber matrix interface. Slowpropagation would be consistent with the observation of multipledirectional changes in the propagated crack as well as multiplebifurcations in the crack patterns. The rapid propagation of cracks inregions containing collagen bundles somewhat perpendicular to thespecimen's length would occur with very low energy absorption. Crackpropagation is expected to be arrested by the presence of canaliculaeand lacunae. In the case where the crack enters a discontinuity, itsfront is blunted, hence reducing the stress concentration factor andtemporarily arresting crack propagation (Piekarsky, 1970). Lacunae areprobably more likely to act as stress concentrators than canaliculaebecause of their ellipsoidal cross section. The orientation of the majorand minor axes relative to the local stress field would determine ifthese structures would likely be the source of fracture initiation.Cracks will initiate sooner in the tissue surrounding the smaller radiuswhen subjected to a local state of tension. Since porosity is expectedto act as a crack arrester, porosity may contribute to increase therobustness of the bone (Currey, 1962). Hydroxyapatite crystallites mayadditionally be pulled out from collagen around canaliculae.

The proposed model simplifies the lamellar structure; in particular itexcludes partially calcified collagen bundles (Ascenzi A. et al., 1965)because their structure is not yet well understood.

Finite Element Model

The Finite Element Method modeling technique proposed here is wellestablished. The model is made on a SP Cluster Machine, which can handlethe extraordinary number of elements. The PI has experience with twoprior micro structural models. The first model consisted of a brightlamellar model (Ascenzi M.-G., 1999) which, in addition to explainingprestress, was able to explain the difference between alternate osteonand lamellar stiffness values found experimentally by independentresearchers (Meunier, 1999). The second model simulated a longitudinalosteon under cyclic torsion (Ascenzi M.-G., 2000). The parametricanalyses proposed in this proposed study will provide the flexibility toinclude lamellar details, that have not been established in theliterature. If the proposed model does not reflect our test resultsrelative to the experimentally observed fracture patterns and overallmechanical behavior, the nature of the differences are more closelyanalyzed in terms of the parametric values. Thus, it will explain therole of the dominating features that control the response to mechanicalloading.

EXAMPLE 7 Thickness Variations Within Dark and Bright Lamellae

For this and the two experiments described below, the source of bonematerial consisted of adult human femoral shafts (27-49 years old), freefrom evident skeletal pathologies, and removed from cadavers inaccordance with US regulations. The selected age range corresponds tothe young age group of Kuo et al. (1998). A rotating-saw microtome witha continuous watering system to provide lubrication and prevent thematerial from overheating was employed to cut longitudinal segmentsapproximately initially 30 mm long from the mid shaft of the femur. Thesegments were then sliced transversely into 70-120 μm thick slabsections. Micro-X-rays of these transverse slabs were prepared to allowidentification (Amprino and Engström, 1952) of fully calcified osteons.

The thickness of the wet outer lamellae of alternate osteons wasmeasured along their two ends, separately, on the thin transverse slabs.The lamellae were characterized as dark or bright by means of polarizedlight microscopy. Dark and bright lamellae show thickness variationbetween the two ends. The result points to the appropriateness ofthickness measurements on lamellar specimens for the proposedinvestigations.

Methods. The lamellar thickness measurements were obtained on slabspecimens with a height of 70±3 μm. Lamellar thickness was measured at 5points on each of the two ends of 86 dark and 66 bright peripherallamellar specimens while still embedded in wet alternate osteons using aDelta Sistemi IAS 2000 image analysis system.

Results. Intra-lamellar standard deviation equals 0.87 μm and 0.88 μmfor dark and bright lamellar specimens, respectively. The followingtable shows means and standard deviations (in microns) between the twolamellar types.

TABLE 5 Thickness On Thickness On Lamellar Type End 1 End 2 ThicknessDifference Dark 7.62 ± 1.91 8.58 ± 1.99 0.48 ± 0.47 Bright 4.27 ± 1.004.85 ± 0.96 0.29 ± 0.26

Differences between bright and dark lamellae were significant (p<0.05).The thickness of both dark and bright lamellar varies between lamellaand within any given lamella along the osteon length as previouslyobserved by Ascenzi A. et al. (1982) on isolated bright lamellae.

EXAMPLE 8 Thickness Variation Between Wet and Dry Lamellar Condtions

Dry and wet lamellar thicknesses were measured within the osteonsamples. Whether dry or wet, dark lamellae are thicker than brightlamellae and bright lamellae expand with water to a lesser degree thanthe dark lamellae.

Methods. The specimens were chosen on 70±3 μm thick transverse sections.Lamellar thickness and width were measured at 5 points on each of 20dark and 20 bright peripheral lamellar specimens in dry alternate osteonspecimens by using a Delta Sistemi IAS 2000 image analysis system, andagain after wetting with distilled water using a micro pipette. On eachlamella, all measurements were taken consistently on one of the twotransverse ends. Only thinner dark lamellae were used for comparisonwith bright lamellae because dark lamellae are in general thicker thanbright ones (see e.g. Marotti et al., 1994), irrespective of whetherthey are dry or wet.

Results. The mean height ± standard deviation of the whole alternateosteon equaled 70.30±9.28 μm and 72.45±9.58 μm, under dry and wetconditions, respectively. The following table shows means and standarddeviations (in microns) on which the student t test was run withsignificance set at 0.05.

TABLE 6 Lamellar Type Thickness Dry Thickness Wet Dark 4.13 ± 1.23 4.10± 1.10 Bright 3.30 ± 0.88 3.56 ± 0.93

Whether dry or wet, bright lamellae are significantly thinner than darklamellae when enclosed in alternate osteons. Additionally, wet and dryconditions affect bright and dark lamellar thickness differently. Brightlamellae are significantly less thick when dry than wet. In contrast,dark lamellae thickness does not change significantly between wet anddry states. The thickness of bright lamellar increases from dry to wetsupporting the hypothesis that bright lamellae contain a higher relativepercentage of mucopolysaccarides, that tend to expand with moisture, andthat the bright lamella tightly encircling dark lamella impedesexpansion. The height of the whole osteon is significantly greater whenwet than when measured in the dry state.

EXAMPLE 9 Isolation of Dark and Bright Lamellar Specimens

This Example describes a new technique which allows isolation of bothdark and bright lamellar specimens from alternate osteons. An alternateosteon specimen is separated from the surrounding transverse bonesection, and the individual coaxial lamellar layers successivelyseparated. Previously, Ascenzi A. et al. (1982) isolated specimens ofdark lamellae by a technique that cannot be reproducibly applied toisolate individual bright lamellae.

Method. Fully calcified alternate osteons were identified on theprepared sections by means of micro X ray and polarized lightmicroscopy. A dental drill secured to a microscope stage cut a trapezoidaround each chosen osteon specimen (FIG. 39). For osteon immobilizationduring lamellar isolation, the base of the trapezoid away from theosteon was glued to a slide. The dark and bright lamellae at theperiphery of each osteon were dissected wet with a razor sharp microtomeblade, obtained by filing a steel needle. This is achieved by firstseparating the interstitial bone outside the alternate osteon from theouter lamella and then separating the outer lamella from the rest of thealternating osteon. The isolation of the single lamellar specimen is avery delicate operation that requires a lot of patience and accuracy.Care is taken so that remnants of interstitial bone and/or of adjacentlamella do not remain attached to the lamellar specimen. Because it isnecessary to secure the bone specimen during isolation, roughly onlyhalf of the lamella can be isolated circumferentially. The lamellarspecimens were then straightened in a ribbon like shape. To avoid thecreation of fractures during straightening of each lamellar specimen,the operation is performed gently on wet specimens while under directmicroscopic visualization. The selection of external lamellae, of lessercurvature than internal lamellae, decreases the risk of fractureformation during flattening. The difficulty of applying our free handmicro dissection technique to obtain regularly dissected anddiscontinuity free specimens is evidenced by the fact that approximatelyonly 1 out 3 specimens successfully complete the procedure. Thistechnique can be performed on transverse sections that measure 70 120 μmin thickness. The lower bound of 70 μm is chosen to ensure flatness ofthe section. The upper bound of 120 μm is chosen to ensure dissectionreliability without indentations. The ribbon specimens are placedbetween two glass plates and wet with distilled water.

EXAMPLE 10 Circularly Polarized Light Microscopy on Dark and BrightLamellae

The birefringence, that manifests itself as bright and dark annularcircles when lamellae are viewed transversely under polarized light, isassociated with collagen bundle arrangement. When isolated, flattenedand wet lamellar specimens are viewed through the lamellar thickness, acontinuous variation in patterning results as the bright lamellarspecimen is gradually rotated through 90 degrees relative to thepolarizing plane. A more discrete patterning variation is observed ondark lamellae examined under the same conditions. This indicated thatthe bundle orientation expresses some variation throughout thethickness. At selected orientations there is a clear domination of apreferred patterning arrangement.

Absence of variation would indicate an amorphous morphology of themicrostructure of the specimen.

Method. 102 dark and 110 bright lamellar specimens were examined with aLaborlux Leitz equipped with polarizing light. A Zeiss laser scanmicroscope equipped with argon ion and helium neon lasers was used toinvestigate the structure and to obtain digital images of the structure,respectively. The images were archived on a hard disk and subsequentlytransferred to film. To increase the digital image fluorescence,structures of interest were stained with a diluted solution of eosin.For comparison purposes, a few 1-2 μm thick sections were cut at aslight angle with respect to the transverse plane from decalcified boneembedded in paraffin. The lamellae appeared better separated than in thecross section and consequently better optical resolution was obtained.

Results. When an isolated, flattened, wet lamellar specimen is observedmicroscopically through its thickness under a polarized light, itsfeatures change in relation to the lamellar orientation with respect tothe polarizing planes of the two Nicol's prisms. When the long edge of adark lamellar specimen is oriented at 0° with respect to the polarizingplane, the light and dark fields align obliquely at ±45°, to the longedge of the specimen. When the long edge of a dark lamellar specimen isoriented at 45° with respect to the polarizing plane, the light and darkfields align perpendicularly to the long edge of the specimen (FIG. 40).

When the long edge of a bright lamellar specimen is oriented at 45° withthe polarizing plane, bright and dark fields align parallel to the longedge of the specimen (FIG. 41). When the long edge of a bright lamellarspecimen is oriented at 0° with the polarizing plane, bright and darkfields align obliquely at ±45°, to the long edge of the specimen.Collagen bundles reveal intermediate orientations at intermediateorientations of the bright lamellar specimen with the polarizing plane.The collagen bundles appear homogeneously distributed along the wholelamella for both lamellar types, except at a few locations. Suchdiscontinuities may really exist or be due to artifactual removal ofbundles during dissection or be due to an optical elision ofsuperimposed orthogonal bundles. The generalized continuous distributionof dark and bright fields as viewed under polarized light in the brightlamella affirms the expectations of various authors (e.g.,Giraud-Guille, 1998) and is consistent with morphological and materialscience points of view.

Additional studies by confocal microscopy and X-ray diffraction suggestthat oblique patterns might be minimal in bright lamellae.

EXAMPLE 11 Confocal Microscopy on Dark and Bright Lamellae

Scanning dark and bright lamellae shows differences in morphology. Sincethe observed morphological details leave room for severalinterpretations, larger specimen groups with finer scanning need to beanalyzed. This confocal microscopy study was motivated by the results ofthe polarized light examination and was conducted to visualize collagenbundle orientation features in three dimensions, without overlap ofcollagen bundles of various orientation and/or ground substance.

Methods. 3 dark and 3 bright lamellar specimens were isolated asdescribed herein. The dimension of greatest length corresponds to thecircumferential direction of the layer in situ. Lamellar specimens werescanned wet every 1 μm in the thickness direction by a Leitz confocalmicroscope. Since the natural fluorescence of wet bone worked well withthe confocal microscope, no staining of the specimen was necessary. Thelight detected by the photomultipliers was converted to pseudocolor inred to improve visualization of the image.

Results. The dark lamellar specimens show a regular arrangement ofunderlying structure across the width of the specimen (FIG. 42). Thebright and dark lamellae revealed different structural compositions. Inthe case of the dark lamellar samples discrete elements of components ofthe microstructure are highlighted. These consist of fibrillar elementsin the plane of the image (lines) and additional elements (dots) alignedperpendicular to the same plane. There is a hint of predominantdirectional orientation of these fibers perpendicular to or at a slightangle to the width of the specimen that corresponds to the overallorientation of the intact osteon specimen. For the bright lamellarsamples (FIG. 43) there are no such distinctly bright elements butrather there are diffuse patches of substance present that are alignedat an orientation of approximately 45° to the prepared specimendimensions. Such patches are believed to be part of a domain. Theabsence of distinct fibrillar structures would support the postulatethat there are differences in structural composition between the twotypes of lamellae. It may also be that the images were not obtained at asufficiently high scanning resolution to be able to clearly discriminatebetween individual collagen bundles. The differences observed in the twotypes of lamellar specimens suggest that they serve distinctly differentmechanical and possibly biological functions. So far we have notobserved in dark lamellae by confocal microscope the oblique collagenbundles observed under polarizing microscope. As stated above, anadditional explanation for the differences observed is that by scanningevery 1 μm in the thickness direction, discrete structures may have beenmissed since the collagen bundle diameters are of the order of 0.5 0.8μm (e.g., Herring, 1972). Examination of more lamellae by confocalmicroscopy in increments of 0.25 μm in the thickness direction isnecessary to resolve this possible deficiency.

EXAMPLE 12 X Ray Diffraction on Dark and Bright Lamellae

Dark and bright lamellar specimens were investigated by Small- andWide-Angle X-ray diffraction (SAXS and WAXS, respectively) for collagenand hydroxyapatite crystallite pattern orientations. These findingsinfer that collagen bundle orientations and hydroxyapatite patternsdiffer between dark and bright lamellae and follow analogous patternswithin the same lamellar type. These results further support Gebhardt'stheory.

Methods. 13 fully calcified dark and 13 bright lamellar specimens wereisolated from 70±3 μm transverse sections and flattened as describedherein. This investigation was designed to check the hypothesis ofcollagen and hydroxyapatite pattern differences between dark and brightlamellae. SAXS and WAXS lend themselves well to such studies becauseSAXS is indicative of collagen orientation through the “staining” of thecollagen bundles by the hydroxyapatite crystallites while the WAXS isindicative of hydroxyapatite pattern orientation (but not theorientation of single crystals). The direction of the incident beam wasdirected parallel to the lamellar plane perpendicularly to the darklamellar specimen length direction to investigate patterns that arepresent and are parallel to the specimen length (FIG. 44). The brightlamellar specimen was rotated by 45° with respect to the incident beamto investigate oblique patterns. SAXS and WAXS diffraction patterns wererecorded using the scanning diffractometry setup of the ID13 microfocusbeamline of the European Synchrotron Radiation Facility (Grenoble,France). The beam wavelength of 0.964 Å was obtained with a Si(111)monochromator and focused to 7 μm (full width at half maximum) by aglass capillary. The sample was scanned through the beam by a computercontrolled x/y gantry. Intact scales, as well as portions of scales weremounted on a goniometric head, with the surface of the scale orthogonalto the X-ray beam. Square areas of up to 50×50 μm were scanned withspatial resolutions of 5 μm and 10 μm. Diffraction patterns wererecorded while moving the sample along the horizontal and vertical axes.Also, some horizontal and vertical linear scans were performed withspatial resolutions from 5 to 20 μm. Small and wide angle patterns wererecorded sequentially on the same area by changing thesample-to-detector distance. The diffraction patterns were recorded witha MARR CCD detector with exposure times of 10 and 30 sec for wide andsmall angle respectively. The detector features are: 2048×2048 pixels,64.45 μm×64.45 μm; 16 bit readout. The thin circumferential hemisectionsand transverse sections of fully calcified alternate osteons wereinvestigated at various angles with respect to the incident beam usedfor verification of results.

Results. An examination of the experimental images shows the following.The SAXS images of dark lamellae are unchanged in shape within scannedareas. This is evidence of a preferential orientation of collagenbundles (FIG. 45). A clear arching of the small-angle meridionalreflection, which corresponds to the third-order collagen periodicity,is indicative of collagen orientation with respect to the osteon axis.The arching shows no change within and across the scanned areas withintensity preferentially distributed in one direction. This isinterpreted as representing a single preferential collagen bundledirection. Moreover, the position of the maximum intensity orientedperpendicularly to the bright lamellar width and the arching of thesmall angle meridional reflection parallel to the bright lamellar widthare indicative of collagen bundle orientation parallel to the osteonaxis (FIG. 46). In particular, X-ray diffraction does exclude largeamounts of collagen bundle and hydroxyapatite orientation oblique to thespecimen length in dark lamellae.

The WAXS images of dark lamellae show a consistent preferentialorientation of the 002 reflection parallel to the bright lamellar width,which indicates one preferential orientation of hydroxyapatite patternsalong the osteon axis (FIG. 47). In contrast, the SAXS images of brightlamellae change in shape (intensity and inclination) within scannedareas, evidencing one or two preferential orientations of collagenbundles at ±45° lamellar width direction (FIG. 48). Therefore, brightlamellae contain areas with oblique collagen bundles, at approximately±45° with the osteon axis. The arching of the SAXS meridional reflectionis unclear on bright lamellar specimens. This is interpreted as a lackof specific collagen orientation. The WAXS images on bright lamellaeshow a preferential orientation of the 002 reflection at 45° withrespect to the lamellar width direction in only some areas (FIG. 50).This indicates the local presence of hydroxyapatite orientation obliqueto the osteon axis. The lack of consistent preferential orientation ofhydroxyapatite patterns suggests a higher directional disorder incomparison with dark lamellae. Note the consistency of the results forcollagen bundles and hydroxyapatite patterns, which support thesimilarity of patterns between collagen and hydroxyapatite. Experimentalimages are currently being examined for differences in hydroxyapatitedensity between the two lamellar types. The clarity of the resultregarding the longitudinal patterns of dark lamellae, shows that theoblique patterns, if present, should be minimum so as not to createenough structural disorder. This result supports the hypothesis thatoblique patterns are not present in dark lamellae.

EXAMPLE 13 Long Bones' Osteonic Lamellae, the Building Blocks of CompactMacrostructure

This Example reports the assessment of the dominant orientations ofcollagen bundles and hydroxyapatite crystallites in dark and brightlamella.

As discussed above, bone is hierarchical, non-homogenous andanisotropic. The building blocks are entities with simple knownstructure, which form a pattern. Starting in the 1950's, AntonioAscenzi's group consistently isolated microscopic units of bone andtested them mechanically. Accurate computer modeling of bone behaviorallows reliable estimates of stresses applied to bone which can beapplied for, e.g., to study fractures; where and how they start andspread. This Example focuses on compact bone: osteons and their coaxiallayers, lamellae.

Osteons are not building blocks: even though they have a simplestructure, they do not form a pattern, whereas lamellae are. Bonemicromechanics finds justification in bone morphology. Back in 1930,Petersen introduced the concept that bone structures can be viewed as afour-order hierarchy arranged in decreasing size. The first ordercomprises the structures corresponding to gross shape anddifferentiation between cancellous and compact bone. The second order ofcompact bone includes haversian systems (or secondary osteons),cylindrical lamellar systems, and additional related structures, e.g.bone marrow. The third order consists of collagen fibrils, which lie inthe ground substance. The fourth order consists of the molecularpatterning between organic and inorganic phases.

Compact adult bone varies in degree of calcification and collagen bundleorientation, and these are both crucial variables in any bone model. Theosteons of compact bone can be studied by micro-X-ray, using amicro-focus and a high resolution film (more than 2000 lines permillimeter), clearly revealing osteons of 220-250 micron diameters. Wetbone, for example shaft transverse sections, under circularly polarizedlight shows dark and bright lamellae.

Many researchers have worked on the question whether collagen bundlesand/or hydroxyapatite crystallites show similar or different densitiesand orientations in dark and bright lamella, starting with Rauber (1873)and then Gebhardt in 1906. Some people hypothesized differences in phaseorientation and other people differences in phase density (Marotti).According to the invention, we have indeed found differences in bothorientation and density.

Lamellae are building blocks of the compact macrostructure. For example,in a femur sample, a 90° rotational pattern from proximal to distalthird of the largest squares, which indicate dominance of brightlamellae, corresponds to the areas of maximum compression. This type ofcorrespondence is present in all long bones, and the bright lamellarpercentage increases with the compressive stress magnitude. FIG. 59shows an example of femur analysis. In this figure, the largest squares,which indicate dominance of bright lamellae, show a clockwise rotationfrom upper to lower third corresponding to the concave regions of thefemur. Since the femur has two curvatures, compression on the femoralhead produces bending. Bending manifests itself in tension on the convexregions and compression on the concave regions. The combination ofbending with axial weight-bearing gives a distribution of compressionmaximum on the concave regions and minimum on convex regions. Otherpatterns are also found that relate the lamellar type to gross geometry.Accordingly, lamellar structure is important for bone modeling.

As previously described, lamellae can be isolated in two ways, one ofwhich applies to both lamellar types. In this method, a trapezoid is cutfrom a thin transverse section and glued to a microscope slide (FIG.39). The outer lamella is then isolated by a free-hand technique, andthe lamella is not isolated in its entirety because of the need to holdthe specimen. After isolation, the lamella is gently flattened (FIG.51). The width (i.e., breadth or length) measures approximately 70microns.

Microscope studies using circularly polarizing light can be done bygradually rotating a wet lamellar specimen on the stage plane through 90degrees relative to the polarizing planes of the two Nicol's prisms.FIG. 40 shows dark lamellar specimen: bidirectional close to ±45 degreesand unidirectional close to longitudinal collagen. FIG. 41 shows brightlamellar specimen: bidirectional ±45 degrees and unidirectionaltransverse collagen. Thus, the main difference between dark and brightlamellae is the presence of longitudinal collagen bundles in darklamellae and of transverse bundles in bright lamellae. Differences amongoblique collagen patterns are not apparent. Moreover, polarizing lightdoes not allow for establishing thickness location of specific collagenbundle orientation.

For further studies of lamellae, X-ray diffraction in the form of SAXSand WAXS can be applied. Collagen orientation is detected through the“staining” of the collagen bundles by the hydroxyapatite crystallites.SAXS and WAX are indicative of collagen and hydroxyapatite orientation.SAXS focuses on the 64 nm collagen 3rd period (see, e.g., Hodge-Petruska(1963) model of the collagen fibril). The 3^(rd) period of collagen isthe groove where the hydroxyapatite crystal sits. Therefore, SAXS isindicative of collagen because the collagen is tinted by thehydroxyapatite. WAXS, on the other hand, is indicative of thehydroxyapatite crystal axis because the unit c-axis coincides with thecrystal long axis. WAXS focuses on the 0.688 period along the c-axis ofthe unit cell Ca₁₀(PO₄)₆(OH)₂. WAXS describes the 002 reflection,wherein 002 refers to the intersections of the plane that describes theperiod with the three axes of reference: In common 3D graphicalrepresentations of crystal patterns, 0 refers to the O-intercept and 2with the ½ the intercept with the c-axis, which coincides with the axisof the crystal.

Lamellar specimens, thin longitudinal hemisections and transversesections of alternate osteons should be positioned normally to theincident beam (FIG. 44). Such sections can be used for lamellar resultverification. As shown in FIG. 45, images are essentially unchangedacross the scanned area. In the enlargement shown in FIG. 46, the cleararching and maximum intensity orientations show single preferentialcollagen bundle direction essentially parallel to dark lamellar width.Also, the diffuse inorganic phase is oriented along the lamellar width,i.e. the osteon and long bone axes. In FIG. 52, SAXS and WAXs analysisof dark lamella shows that collagen bundles and adjacent hydroxyapatitecrystallites follow the same orientation. FIG. 53 shows bright lamella,wherein the images change across the scanned area (details in FIGS. 54and 55).

FIGS. 56 and 57 show the results of X-ray diffraction data, revealingthe dominant orientations of dark and bright lamella. Specifically, thehighest percentage of collagen bundle and hydroxyapatite crystallitesorientation is in a single range, that between 67.5 and 112.5 (90±22.5)degrees. In bright lamella, there are two orientations, so that thecollagen bundle and hydroxyapatite crystallites orientation rangesbetween 22.5 to 67.5 (45±22.5) degrees, and between −67.5 to −22.5(−45±22.5) degrees, wherein the lamellar length is parallel to thex-axis. For the bone model of the invention, this means that whenlamella and their collagen bundles and hydroxyapatite crystals areincluded in the model, the dominant percentages should average betweenthese values.

FIG. 58 shows how the different orientations can be modeled, e.g., usinga twisted plywood structure.

These above findings can all be incorporated into the bone modelingconcept, as well as the dimensions of the domains, collagen densitywithin domains (which can be studied by confocal microscopy) and themechanical properties of lamellae (which can be studied by tensionalloading with a microextensometer).

EXAMPLE 14 Structural Dimensions and Patterns Within Single SecondaryOsteons

This Example explores the structural dimensions and patterns withinsingle secondary osteons, with consideration of their biologicalvariation. Stereometric data, combined with data from examination byregular and circularly polarized light, confocal microscopy, synchrotronX-ray diffraction and micro-focus high-resolution micro-X-ray, providethe basis for models, preferably computerized, of single osteons andsingle lamellae. These models provide the concurrent representation of(1) collagen-hydroxyapatite orientation, (2) relative hydroxyapatitepercentage, (3) distributions of osteocytes' lacunae and canaliculae,and (4) biological variations in the dimensions of the relevantstructures. The mathematical software Maple™ can be employed toimplement the computerized models. While the parts of the models arepreferably constructed on a personal computer, the voluminous dataassociated with the representation of lacunar and canaliculardistributions in this Example require a supercomputer for the assemblyof the models and final analysis. The programming used to define themodel affords the option to randomize the dimensional specifications ofosteons, lamellae, lacunae and canaliculae within the experimentallyobserved numeric ranges and distributions. Through this option, theprogram can operate so that each run of the file produces a uniquerandom model within the observed biological variations. Further, theprogram can also be run to implement specific dimensional requirements.The modeling has applications in the micro-structural study of fracturepropagation and remodeling, and in the simulation of mechanical testing.

This Example focuses on identifying and representing the structuralpatterns that are observed within a single secondary osteon, as anexpression of the biological variations within the Haversian system. Thecomprehensive identification of the patterns comes together in acomputerized modeling methodology that, within the experimentallyobserved biological variations, allows the structural simulation of asingle osteon and/or a single lamella.

Two interconnected motivations justify the attention to osteons. First,the complexity of anisotropy and non-homogeneity in the macrostructureof compact bone can only be unraveled by descending to the osteon leveland identifying the osteon's lamellae as building blocks of themacrostructure (Ascenzi A., 1988). Second, important current biologicalproblems, such as transduction mechanism appraisal, fracture riskassessment, implant success prediction, and evaluation of new therapiesfor bone metabolic and remodeling disorders, depend on understanding thelocal microstructural properties around bone cells.

The structure of secondary osteons has remained an open question sincetheir first microscopic observation (Leeuwenhoek, 1693). Numerousmicroscopic techniques (e.g. Ebner, 1875; Kölliker, 1854; Ranvier, 1887;Gebhardt, 1906; Ziegler, 1908; Weidenreich, 1930; Amprino, 1946; Ruth,1947; Rouiller et al., 1952; Engström and Engfeldt, 1953; Frank et al.,1955; Frank, 1957; Vincent, 1957; Smith, 1960; Currey, 1964; Ascenzi A.and Bonucci, 1967, 1968; Boyde, 1969; Marotti, 1979; Reid, 1986;Giraud-Guille, 1988; Ascenzi A. et al. 1987, 1994, 1997; Ascenzi M.-G.,1999; Carter and Hays, 1977; Ascenzi M.-G. et al., 2003) have beenemployed towards the recognition of the relevant structural variablesand the generation of appropriate models. Collagen fibril orientation,degree of calcification, and distribution of osteocyte lacunae andcanaliculae are relevant variables. The range of collagen densityvariability and its relevance remain unclear. The most recent theory onosteon's lamellar structure (Ascenzi M.-G. et al., 2003) differentiatesthe lamellar types in terms of collagen bundle and hydroxyapatiteorientation and is based on findings by regular and circularly polarizedlight, confocal microscopy, synchrotron X-ray diffraction andmicro-focus high-resolution micro-X-ray on isolated osteon lamellae.

The methodology of this Example starts with the expression of thegeometry of a single osteon or a single lamella by means of thick hollowcylinders, cylindrical shells, ellipsoids and thin long cylinders(Andreuzzi, 2003). It expands the existing structural theory regardingthe collagen-hydroxyapatite distribution within the osteon to includelacunar and canalicular distributions. The methodology is checkedthrough comparison of the generated model with an actual specimen. Newinformation emerges on the coexisting distributions of lamellar, lacunarand canalicular distributions across the spectrum of osteon types.

Method

This Example presents a new methodology to model the biologicalvariability of compact bone microstructures. FIG. 60 shows examples ofsuch variability. The new methodology is based on the identification ofexperimentally observed structural patterns. It is independent of theparticular experimental data that is employed to define the ranges ofvariability in the model. Experimental data on osteons presented here isemployed to verify the model.

Experimental Data

The osteon data is obtained from the observation and measurement ofphotographs taken under regular and circularly polarized lightmicroscopy of 124 transverse sections and of 54 longitudinal sections offemoral osteons at magnifications ranging between 190 and 600. Suchphotographs were prepared by Antonio Ascenzi and Alessandro Benvenuti atthe University of Rome “La Sapienza” during their decades of research onosteons.

The photographic material was prepared as follows. Post mortem femoralshafts, free of evident bone pathology, of male humans aged 19-40,provided the bone material. The middle shaft of each femur was removed,defleshed and air-dried. A rotating saw microtome equipped with acontinuous water lavage to prevent heating of the material was used tosection the material. The middle-shafts were first cut into 3 cm longtransverse sections. The 3 cm long transverse sections were cut into50-100 μm thick either transverse or longitudinal sections. All thesections were micro-X-rayed (Amprino and Engström, 1952) to assess thedegree of calcification of the osteons. The micro-X-rays were examinedunder regular light to select osteons that were mostly cylindrical inshape and at either the beginning (70%) or final stages (100%) ofcalcification. The sections were then examined under circular polarizinglight to choose the subset of osteons, which are representative of thespectrum of all possible lamellar arrangements. Such osteons are thosethat under circularly polarizing light in transverse section appearpredominantly dark (extinct osteons), alternatively dark and bright(alternate osteons), and predominantly bright (bright osteons). Thelongitudinal sections were then used to prepare osteon specimens bylatheing. A Leitz camera was used to photograph each of the osteonschosen on the transverse and longitudinal sections, perpendicularly toeither the section's transverse cut or the osteon lathed surface. Thefocus of the camera was adjusted so only the generally longitudinallyoriented lacunae appear on the longitudinal section.

A precision scale was used to measure the details of the osteonstructure shown on the photographs. The dimensions of larger entities,such as osteon radii, were measured in triplicate, while the dimensionsof smaller entities, such as lacunae and canaliculae, were measuredtwice by two different technicians to ensure measurement consistency.The error was at most on the order of ±1.31 μm. Osteon radii, number oflacunae, lacunar dimensions and relative distances, and canalicularlength were measured on photographs of transverse sections. Lacunardimensions and relative distances were measured on photographs of osteonspecimens obtained from longitudinal sections. Tables 7 and 8 presentthe ranges and averages of each measured entity across the five osteongroups: extinct osteons at final stages of calcification, alternateosteons at either initial or final stages of calcification, and brightosteons at final stages of calcification. The extinct osteons at initialstages of calcifications were too few to be included.

Structural Patterns

The following patterns emerged from observation of the photographs:

-   -   I. The lacunae closer to the cement line, that is the osteon        outer lateral boundary, lie on the outer-most one or two        lamellae and are among the largest present in the osteon        section.    -   II. The lacunae closer to the Haversian canal lie on the        inner-most one or two lamellae.    -   III. Any three adjacent lacunae on a transverse section are        located at a staggered distance from the Haversian canal and can        be visualized as located at the vertices of a triangle.    -   IV. The lacunar cross section is rounder and less elongated in        extinct osteons than in alternate and bright osteons.    -   V. The canaliculae are mostly generally radially distributed on        planes transverse to the osteon axis. They appear to spread as        their distance from the lacuna where they initiated increases.        This is especially pronounced for the canaliculae running        towards the cement line.

Mathematical Model

The model is developed from the above-identified patterns and fromresults as to collagen bundle and hydroxyapatite orientation (Ascenzi,M.-G. et al., 2003) as explained below.

The construction of the model is realized with a computer programcontaining procedures of Maple™ (Waterloo Maple Inc., Waterloo, Ontario,Canada;) syntax. The program provides for the construction of athree-dimensional model. The program begins with the option to buildeither a random or a specific model or a combination of the two. If therandom model is chosen, at any time during the model's construction whenan entity dimension needs to be entered, the program chooses a randomvalue within the entity's range (from Tables 7 and 8) and along itsobserved distribution. The tabular data can be replaced by any otherexperimental data set of osteons on transverse and on longitudinalsections. If the specific model is chosen, the user specifies thedesired dimensional choices. This second option was introduced to testthe program's flexibility and ability to reproduce the photographicimages of osteon specimens. The third option combines the previous twoto allow certain dimensions to be chosen randomly while others arespecified. This option serves to build models of specimens where onlysome of the constituent dimensions are known. The following explanationrefers to the random model because it is more general and any randomchoice can be replaced by a user's choice.

Once the osteon type (extinct, alternate or bright) is chosen, theprogram builds the model's geometry, which is differentiated betweenglobal (model of single osteon with lamellae or single lamella) andlocal (model of lacunae and canaliculae). The global geometry consistsof thick cylinders and cylindrical shells, which model osteons andlamellae (Ascenzi M.-G., 1999), respectively. Within the globalgeometry, the description of the collagen and hydroxyapatite orientationis specified. The local geometry consists of ellipsoids that modellacunae (Ebner, 1875) and thin cylinders that model canaliculae. Boththe global and local geometry are specified in terms of equationsobtained by appropriate translations, rotations and intersections ofcylinders and ellipsoids initially centered at the origin of a referencesystem. Because the lamellar model is obtained from the osteon model bya mathematical process that simulates the experimental isolation of asingle lamella portion from an alternate osteon (Ascenzi, M.-G. et al.,2003), the osteon model is presented first.

Global Geometry

The random choices of external and internal diameters and height for theosteon model specify the global geometry's hollow cylinder. The sequenceof the osteon's lamellae is then modeled in dependence of the osteontype (extinct, alternate or bright). If the alternate osteon type ischosen, either the program chooses randomly, or the user specifies,whether the lamella adjacent to the Haversian canal is extinct orbright. The thickness of the thin coaxial cylinders that model thelamellae are then chosen randomly or by the user within the rangesobserved experimentally under circularly polarized light during theyears of experience with lamellar isolation: 3-15 μm for extinct and2-12 μm for bright lamellae. Such values can be replaced by the user'sdesired values. The procedure for selecting the random lamellarthickness values stops as soon as the osteon model thickness is achievedor exceeded. If exceeded, the last lamellar thickness is excluded andthe difference between the osteon model thickness and the sum of thelamellar models' thickness values is randomly divided and distributedamong the lamellar models.

Each extinct and bright lamellar representation in the model is thenassociated to a function that represents the collagen and hydroxyapatiteorientation distribution with respect to the osteon model axis. Theunidirectional dominance at 0 degrees within a [−45,45] degree range forthe extinct lamella and a bi-directional dominance at ±45 degrees withina [45,135] degree range for the bright lamella (FIG. 61 a) by X-raydiffraction is translated mathematically as follows. Thecollagen-hydroxyapatite orientation is modeled by continuous and smoothfunctions in agreement with the continuous and smooth variations (FIG.61 b) observed by polarized light on lamellar flattened specimens.Because X-ray diffraction does not quantify the “dominance”, in firstapproximation dominance is translated into 75% of the lamellar thicknessfor both lamellar types. Since the “dominance” location is stillexperimentally unclear within the lamellar thickness, the dominantorientation is placed in the middle of the extinct lamella and towardsthe edges of the bright lamella for simplicity. A more precise andsophisticated model of collagen-hydroxyapatite orientation awaitsfurther information from a confocal microscopy study, which is currentlyunderway. FIG. 61 shows the collagen-hydroxyapatite orientation on twoadjacent lamellae of an example of alternate osteon model.

Local Geometry: Ellipsoids

The first step is assignment of the distribution of the centers of theellipsoids. In accordance with the fact that the osteon tissue isgenerated starting from the cement line inward, the centers of theellipsoids are chosen from the osteon model outer boundary inward.Because of pattern I the distances of the centers of the ellipsoidsmodeling the outermost lacunae are chosen randomly so that the centersfall within the two most-outer thin coaxial cylinders modeling thelamellae. By drawing rays from each outer lacuna to an estimated centerof the Haversian canal on the photographs, it is observed that each rayintersected some lacunae and a few more lacunae lay at a relativelysmall distance from each ray. Therefore, a random number of rays (seesolid rays in FIG. 62 a) are chosen corresponding to a fixed angle(22.5° in the figure). The ellipsoid centers were placed along rays at arandomly chosen fixed radial distance from each other within themeasured range starting with one within the two most external thincylinders and finishing with one within the two most internal thincylinders (in agreement with pattern II). Finally, the ellipsoid centerswere translated a random distance from the ray circumferentially withinan ad hoc angular range to create disorder (FIG. 62 b). Unwantedintersections between any two ellipsoids are avoided in two ways: first,by choosing an angular range smaller than the angle between twosuccessive black rays by the angle corresponding to the maximum value ofthe ellipsoid axis; and secondly, by choosing alternatively on any otherray the first ellipsoid's center to be located within the first-secondthin cylinders closer to the model's inner cylindrical surface or withinthe second-third thin cylinders closer to the model's inner cylindricalsurface.

Pattern III was satisfied as the centers of the ellipsoids are placedrandomly along the rays. A random value is chosen for the ellipsoidmajor axis within the range of the major axis of the lacunarlongitudinal section in Table 8 because the exact orientation of thealmost longitudinally oriented lacunae on the photographs is unknown.The medium and minor axis of the ellipsoid are computed by multiplyingthe value of the major axis by a random value within the ranges for themedium-major and minor-medium ratios, respectively. If the found valueis not in the appropriate range, the range value closer to it will beassigned. Such ratio ranges equal [3/4,1] and [1/10,1/2], respectively,and they were found through the process of reproducing osteon imageswith the presented method. The program allows selection of theorientation of the ellipsoid along the collagen-hydroxyapatiteorientation at the center of the ellipsoid or along the osteon axis.This option was introduced to check which is more realistic in thereproduction of osteon photographs by this method. The program allowsthe choice of generating ellipsoids whose transverse sectionconsistently increases with the center's distance from the model'sinternal boundary (Ardizzoni, 2001) or not (the existence of suchpattern has not yet been checked).

From Whole Transverse Section to Sector

The generation of canaliculae relative to the 360° transverse section,even if the height of the model is kept small, would create anunmanageably large file impossible to import into a finite elementprogram. The maximum file size limit is on the order of 15 Mb. To limitthe size of the model's file, the modeling was restricted to a sector ofthe 360° transverse section. To generate the 360° model of desiredheight, the modeled sector is rotated and translated relative to itsaxis after importing it into a supercomputer. To define the localgeometry at the boundaries of the model's sector, symmetries that do notexist in reality are introduced in the model so that the details of thelocal geometry can recombine and provide continuity within the finalmodel.

The osteon sector to be modeled will be a cylindrical sector (see e.g.FIG. 63) corresponding to the angle φ subtended by the minimumcircumferential distance between adjacent, more external lacunae. Forthe experimental data presented here, such angle is on the order of22.5°. The sector will contain three families of ellipsoids whosecenters are initially placed along the initial, middle, and final ray ofthe angle φ. Since the rotated sector will need to recombine with theinitial one, the centers along the initial and final rays will need tobe positioned at corresponding distances from the model's innerboundary. Afterwards the ellipsoids will be moved randomlycircumferentially as in the 360° case, except that here thecorresponding ellipsoid on the initial and final ray of the sector willneed to move by the same amount. In particular, only portions of theellipsoids on the initial and final rays which fall within the sectorare modeled.

Local Geometry: Canaliculae

The radii of canaliculae are not part of the experimental data. Theminimum and maximum values for the canalicular radius are 0.3 and 0.6microns, respectively (Piekarski, 1976), and the random choices areassumed to follow a uniform distribution for lack of furtherinformation.

To generate the thin cylinders that model the generally radiallyoriented canaliculae (pattern V), the coordinates of the pointsdistributed at the vertices of a grid of side d are identified in theellipsoid on the plane spanned by the major and medium axes of theellipsoids. The value of d is chosen larger than twice the maximumdiameter of a canalicula to avoid non empty intersections of adjacentthin cylinders. The program offers the option of chosing a random arandom subset of such points. For each ellipsoid, the thin cylindersstarting at each of the points are then differentiated between the onesthat run from the ellipsoid towards either the outer or inner curvedboundary of the model. For both types, the angle between the thincylinder axis with the initial axis of φ is a random value thatincreases with the distance of the initial point from the center of theellipsoid. This is to allow the spread of the thin cylinders exiting thesame ellipsoid. A range between ±34° yields realistic orientations. Theprogram has the option to interrupt the thin cylinders from running fortheir random length if they intersect ellipsoids different from theellipsoid at which they initiated. When a thin cylinder starts at apoint located along the initial or final ray of the sector only the halfof the thin cylinder that lies in the sector is drawn so that boundariesmatch when the sector is replicated.

While photographs indicate that the majority of canaliculae aregenerally oriented radially and perpendicularly to the osteon canal(perhaps 85%), the rest of the canaliculae that initiate or end atlacunae are modeled as oriented either along the collagen-hydroxyapatiteorientation or along the osteon axis or transversely, as is sometimesobserved on photographs. Further, only the canaliculae that initiate atlacunae are modelled here because a criteria to establish the positionof others is not formulated.

Lamellar Isolation Simulation

The whole outermost lamellar portion simulated in the osteon model by athin cylindrical layer is “isolated” from the model (FIG. 65) tosimulate experimental isolation of outermost lamella from alternateosteon. The ellipsoids and canaliculae of the alternate osteon model(FIG. 65 a) are intersected with the boundaries of the thin layer (FIG.65 b). Since intersection of geometrical objects is a time-consumingcomputer process, the number of geometric objects to be intersected isdecreased by a program procedure that shortens the thin cylinderrepresenting the canalicula to the thin layer representing the lamellarportion by means of diminishing the size of parameters' ranges.

Comparison Between Methodology Result and Experimental Findings

The flexibility and validity of the methodology were evaluated by usingit to reproduce osteon section photographs. For any transverse orlongitudinal osteon section that was modeled by the described method,good agreement was found between photograph and model. That is, if themodel is printed on transparency film at the same magnification of thephotograph, superimposition of model with photograph shows coincidenceof structural entities.

FIGS. 63 and 64 illustrate examples of modeling. FIG. 63 a shows aportion of bright osteon viewed transversely under polarizing light.FIG. 63 b shows the lacunar-canalicular model of an osteon sector. FIG.63 c shows a three-dimensional detail of the sector model with a planeindicating the section represented in FIG. 63 b. The (two) lacunae andtheir canaliculae that intersect the plane appear also in FIG. 63 bwhile the rest of the structure is randomly generated.

FIG. 64 a is the photograph of a portion of alternating osteon viewedlongitudinally under polarizing light. The alternating extinct andbright lamellae and lacunae are visible. The structure delimited by thesquare is simulated is FIGS. 64 b and 64 c. Because the program modelsthe lamellae without their wobbling (Ascenzi A. et al, 1982), thelamellae and their model coincide only at the base of the square. Thelamella-lacunar model in FIG. 64 c provides a representation of thethree lacunae inside the square.

Application of the Mode 1

This Example addresses the question of structural patterns andbiological variation within extinct, alternate, and bright osteons inrelation to collagen orientation, degree of calcification, anddistribution of lamellae, canaliculae and lacunae. The comprehensiverepresentation of these osteon features obtained by the methodologypresented allows the elucidation of hypotheses regarding thelong-standing questions as to osteon structure.

Parameters for the preferred method are the exclusion of (a) fineporosity below canalicular size, (b) interface specification betweencollagen and hydroxyapatite crystallites and (c) domains smaller thanlamellar size. Further, the canalicular axis is modeled as a line andnot a curve as it actually is. Such limitations may be removed at alater date. The accuracy of the method was checked by creating models ofspecific configurations within and at the extremes of the measurementranges. This was made possible by the flexibility of the method, whichby means of parameters allows the user to change the properties of themodel locally. The process of checking the accuracy has proven itselfvaluable in the determination of parameter value ranges to be used inthe preparation of random models, such as density and orientation ofcanaliculae. Because the precision of the model depends on how well themeasurements obtained for the microstructural elements and patternsrepresent the biological variation, currently available photographs ofmicrostructures are being observed and additional ones will be taken toidentify possible additional patterns or dependencies.

Secondary osteon three-dimensional modeling has been viewed in thecontext of the compact bone macrostructure (see e.g. Katz, 1981; Hogan,1992; Aoubiza et al., 1996) or as a self-contained structure (Pidapartiand Burr, 1992; Ascenzi, M.-G., 1999; Andreuzzi, 2003). In the contextof the compact bone macrostructure, the osteon as been modeled as atwo-phase collagen/hydroxyapatite fiber reinforced composite and thenhomogenized into a homogeneous unit with isotropic, orthotropic ortransversely isotropic mechanical properties. Pidaparti and Burr (1992)and Aoubiza et al. (1996) have modeled the osteonic lamellae at a fixedcollagen orientation to estimate the mechanical properties of theosteon. Ascenzi, M.-G. (1999) introduced the variability of collagenorientation through the lamellar thickness in a three-dimensional modelof a single bright lamella for prestress estimate. The same approach wasthen extended in Andreuzzi (2003) to a single osteon model of extinctand bright type. Various authors have investigated the effect of thepresence of Haversian canals, lacunae and canaliculae on osteon tissue'selastic properties (e.g. Cowin, 1999; Sevostianov and Kachanov, 2000).The present example includes both (i) a more realistic continuouslyvariable orientation distribution of collagen across the lamellarthickness in accordance with the lamellar type as viewed undercircularly polarized microscopy and (2) a geometrical representation oflacunae and canaliculae.

The data collected to check the methodology is part of an expandingdatabase. The number of specimens for each osteon group needs toincrease before comparison among the measurements of the various osteongroups is made. Nevertheless, the current findings are within the rangesreported in the literature. The measurements for the major and minoraxes of the lacunar transverse sections are consistent with thecollagen-hydroxyapatite simulation angle in the model. That is, theextinct osteons, whose collagen-hydroxyapatite orientation has beenhypothesized (Ascenzi, M.-G. et al., 2003) as dominantly oriented alongthe osteon axis, show a smaller minimum value of the angle range thanalternate and bright osteons where the collagen-hydoxyapatiteorientation is hypothesized to be dominant transversely with respect tothe osteon axis. FIG. 66 shows how the transverse section of anellipsoid varies with the inclination of the major axis of the ellipsoidwith respect to the transverse plane. The ellipse intersection issmaller and rounder at 90 degrees and it looses the roundness andelongates as the angle decreases to 0 degrees. This fact and the abilityof our methodology to reproduce lacunae in photographic images by meansof ellipsoids whose major axis is aligned with thecollagen-hydroxyapatite orientation is consistent with the assumptionthat lacunae elongation parallels the direction of collagen orientation.The microstructural patterns identified in this example have beenpartially observed previously by already cited authors. In particular,the larger size of the most external lacunae has been previouslyexplained in terms of the higher deposition rate of the relative bonecell at that location (Boyde, 1969). Further, it should be noted herethat the photographs of both transverse and longitudinal osteon sectionsshow canaliculae that suddenly stop. Such stopping could be due eitherto the fact that the canaliculae at that point where they appear to stopcontinue to proceed in the structure out of the plane of focus or thatthey thin down and become invisible for the current magnification. Suchfeature is replicated by the modeling method.

The strength of the proposed methodology lies in its flexibility. First,any of the methodology procedures can be rendered more sophisticated andadjusted to future findings. Second, the methodology relies onparameters that can be varied to match future biological observation ofspecimens. The efficiency of the method in terms of computer time, whichaverages a few minutes on a PC for a sector of approximately 15 μmheight, relies on a balance between the use of functions and sequences.Employment of functions or sequences exclusively maximizes computationaltime. Therefore, function evaluations are computed as numeric listsperiodically in the program to improve the efficiency of thecomputations. Further, mathematical elegance and computer efficiency areoften at odds in the execution of the program. In fact, mathematicalconciseness and simplicity may cost extra computer time, which isunaffordable in this program in view of the voluminous data treated.

An application in progress employs the model's geometry and theestimation of prestress in the bright lamella (Ascenzi, M.-G., 1999) tosimulate osteon cyclic torsional loading by finite element analysis. TheMaple™ image of an alternate osteon sector that contains the informationof collagen and hydroxyapatite orientation and porosity is exported fromMaple™ as a dxf file. The dxf file format was chosen because it retainsthe list of the points' coordinates that specify the geometry. The dxffile was then converted into an igs file, which can be imported into thefinite element software Abaqus™. Abaqus™ is used to rotate and translatethe sector to build a portion of model of an osteon or lamellarspecimen. In order to maintain the total number of elements within thelimitations of the software, it is necessary to work on a homogeneousmodel and then refine the mesh locally to represent lacunae andcanaliculae only in the model's portion of interest by means of themodeling methodology presented here.

A further application of the models is in the investigation of modelingthe mechanism by which fractures initiate and propagate. In fact,material science posits that (1) a fracture initiator is an 8-10 μm longmicro-crack; (2) the orientation of the microcrack has an effect on theminimum stress necessary to elongate the crack at its weakest point, thetip; and (3) discontinuity in the structure as small as a micron can actas crack arresters. Therefore, because of the size match, lacunae arethe obvious candidates as fracture initiators and both lacunae andcanaliculae may act as crack arresters. Since the osteon modelsgenerated by the presented methodology include the representation ofcollagen, hydroxyapatite, lacunae and canaliculae, such models aresuitable to be used to explain the process of fracture initiation,coalescence, and spread. Further, the orientation of the lacuna withrespect to the osteon axis and the osteon's degree of calcificationwould affect the minimum stress necessary to initiate fracture of thelacunae at one of its apices. In particular, since the methodology ofthe invention supports the hypothesis that the lacunar orientationfollows the collagen-hydroxyapatite orientation, the simulated fracturepatterns would be expected to differ among osteons with differentcollagen-hydroxyapatite orientation distributions, as have been observedexperimentally (see e.g. Ascenzi A. et al., 1994). A deeperunderstanding of the factors that control fracture initiation and spreadwould then shed light on the process of fracture propagation in the bonemacrostructure.

TABLE 7 Dimensional measurements of osteons, lacunae, and canaliculae ontransverse osteon sections. Osteon type (% calcification): ExtinctAlternate Alternate Bright Bright (100%) (70%) (100%) (70%) (100%)Number of specimens 36 24 31 20 13 Range Range Range Range Range Entitymeasured (mean ± sd) (mean ± sd) (mean ± sd) (mean ± sd) (mean ± sd)osteon external radius, 84.27-206.38 106.45-163.33 93.75-163.21102.35-153.71 127.95-178.57 μm (121.56 ± 23.50) (137.42 ± 14.44) (123.82± 16.09) (128.27 ± 13.24) (147.88 ± 15.72) osteon internal radius, μm14.11-37.82 25.00-56.31 10.84-41.02 24.33-57.42 10.77-46.77 (25.18 ±6.31) (39.00 ± 6.97) (25.18 ± 8.49) (42.31 ± 8.63) (28.42 ± 12.94)number lacunae 8-56 13-49 14-49 18-54 29-69 (27 ± 11) (29 ± 9) (29 ± 9)(36 ± 11) (51 ± 13) major axis lacunar 1.85-29.79 3.27-29.80 3.03-19.093.19-31.80 3.37-30.77 section, μm (11.29 ± 1.01) (12.81 ± 1.80) (11.20 ±1.41) (12.91 ± 1.63) (14.83 ± 20.74) minor axis lacunar 1.87-12..311.63-10.68 1.04-9.19 3.13-11.28 3.08-10.87 section, μm (5.83 ± 1.10)(5.61 ± 1.32) (6.31 ± 1.31) (5.74 ± 0.84) (5.43 ± 0.83) minimum radialdistance 3.08-13.33 3.23-17.06 3.03-10.60 3.13-10.07 3.03-10.60 betweenlacunae, μm (4.81 ± 2.55) (5.84 ± 3.64) (6.27 ± 2.89) (4.96 ± 2.43)(6.02 ± 2.48) minimum circumferential 3.08-18.45 3.23-17.06 1.64-32.973.19-14.13 3.51-14.49 distance between lacunae, (7.26 ± 4.13) (7.04 ±3.79) (8.21 ± 7.17) (6.07 ± 3.37) (6.54 ± 2.99) μm maximum canalicular9.46-51.06 16.13-86.81 12.99-51.81 16.45-48.87 10.10-53.38 length, μm(26.61 ± 10.75) (31.29 ± 20.36) (27.22 ± 12.05) (27.56 ± 10.40) (25.94 ±12.40)

TABLE 8 Dimensional measurements relative to lacunae on longitudinalosteon sections. Alternate osteon at 100% calcification on longitudinalsections Number of specimens 54 Entity measured Range (mean ± sd) majoraxis lacunar section, μm 6.07-25.00 (20.03 ± 9.01) minor axis lacunarsection, μm  1.57-5.31 (1.99 ± 0.82) minimum distance between 3.66-36.40(20.43 ± 8.60) lacunae, μm

EXAMPLE 15 Confocal Microscopy Experiments

A. Distribution of Collagen Bundle Orientation in Human SecondaryOsteons

Circularly polarized light microscopy, micro-X-ray and confocalmicroscopy allow observation of distribution of domains throughoutisolated lamellar specimens of fully calcified secondary osteons.(Domains are 3D regions of essentially unidirectional collagenorientation. (Boyd (1972)). This approach is novel in its assessment ofdomain characteristics along lamellar thickness direction, i.e., theradial direction, in embedding osteons prior to specimen isolation.Differences in domain distribution are found between the two buildingblocks of osteons: lamellae that appear extinct and lamellae that appearbright in cross-sections under circularly polarized light.

Material: 70±3 μm thick cross-sections were prepared from femoralmid-shafts, aged 19-40, free of skeletal pathologies. Sections weremicro-X-rayed with resolution of 2000 lines per millimeter. Fullycalcified osteons that appear alternatively extinct and bright undercircularly polarized light were identified for lamellar specimenisolation. 18 extinct and 13 bright lamellar specimens adjacent to thecement lines were isolated, Ascenzi et al. (2003), and gently flattenedalong specimen thickness direction.

Methods: Specimens were scanned every 0.5 μm by a Leica TCS-SP confocalmicroscope (Heidelberg, Germany) with krypton laser (568 nm excitation)and 63×Planapochromat lens. The 0.51 μm interval is chosen in relationto plane of focus thickness to avoid either missing, or replicating,structures. Natural fluorescence of wet bone provides good contrastbetween tissue components. Light detected by photomultipliers convertsto pseudo-color for good visualization. Images were digitally memorizedand further magnified and analyzed using Image-I (Sun Microsystems Inc)at 1575× life size (FIG. 1). Bright elements are interpreted as collagenbundles on a background interpreted as a mix of mucopolysaccharides andglycoproteins. Collagen bundle orientation was measured relative to thespecimen length, i.e., relative to a plane perpendicular to originalosteon axis. Domains were identified as continuous regions of at least 3or 4 unidirectional collagen bundles whose orientation varies up to±7.5° both on the plane of focus and across adjacent images.

Results: The bright lamellar specimens show higher variation in collagenbundle orientation distribution in terms of higher number and smallersize of domains and higher number of distinct domain orientations. Whileextinct specimens may show only a domain orientation per area of focus,the bright specimens can show up to 4 domains per area of focus. Thedomain dimensions range between 7.61 and 175.251 μm for extinct and9.27-163.40 μm for bright specimens on the plane of focus. The depth ofthe domain perpendicular to the plane of focus is smaller than lamellarthickness. Depth range is estimated as 1.33-4.38 μm for extinct and1.67-5.00 μm for bright specimens. The measured domain dimensions arelarger than values relative to domains observed by scanning electronmicroscopy near the Haversian canal (Boyde (1972)). The difference isexplained in terms of local stress concentration effects around thecanal which increase collagen orientation variability. The range ofdomain orientations per lamellar type is smaller (12°-158°) for extinctthan (1°-179°) for bright specimens. Consequently, collagen thatparallels specimen length is characteristic of bright specimens. FIG.67( b). The results agree with observations by polarized light andsynchrotron X-ray diffraction (Ascenzi et al. (2003)). Dots on images ofboth specimen types are portions of collagen cut during lamellarisolation from radially oriented collagen in original osteons. FIG. 67(a, b). The findings are in line with the mechanical significance ofcollagen orientations in long bone shafts, Ascenzi (1988), and theskeletal adaptation to mechanical function, Frost (1987).

B. Collagen Bundle Distribution Along Human Secondary Osteons' RadialDirection

Collagen orientation distribution within so-called bright and extinctlamellae, the building blocks of secondary osteons, is observed alongthe previously unexplored osteon radial direction.

70±3 μm thick cross-sections were cut from femoral mid-shafts, aged19-40, pathology-free. 22 fully-calcified lamellar specimens, 11 pertype identified as bright or extinct on transverse section undercircularly polarized light, were then isolated and gently flattened.Ascenzi, M-G (2003). A Leica TCS-SP confocal microscope (krypton laser,63× Planapochromat lens) scanned wet specimens each 0.5 μm throughthickness. Natural fluorescence shows as green pseudo-color at 1575× and3000× magnification with Image-I and Xarax-X. Collagen bundles appear asbright elements on dark background of mucopolysaccharides andglycoproteins. FIG. 68( a). Long portions of canaliculae appear tofollow local collagen orientation. Domains were identified as 3D regionswhose collagen bundle orientation varies to ±7.5° within and acrossadjacent scans. Collagen orientation within domain was measured relativeto original osteon axis.

Bright lamellae show characteristic collagen distribution somewhattransverse to the original osteon axis within the middle thickness, FIG.68( d), becoming gradually oblique towards approximately ±45° relativeto the osteon axis. FIG. 68( b), (c) (d). Extinct lamellae show acharacteristic collagen distribution whose orientation makes up to a±22.5° angle relative to the original osteon longitudinal axis. FIG. 69.

Results conform to findings by different methods. Ascenzi M-G (2003).These include mechanical significance of bone shaft collagenorientations, Ascenzi A (1988), and skeletal adaptation to mechanicalfunction, Frost (1987). Results may be integrated in high-accuracy,super-computer hierarchical bone models, Ascenzi M-G (2004), e.g. totest hip prostheses virtually.

The patents, applications, test methods, and publications mentionedherein are hereby incorporated by reference in their entirety.

Many variations of the present invention will suggest themselves tothose skilled in the art in light of the above detailed description. Allsuch obvious variations are within the full intended scope of theappended claims.

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1. A method of predicting a mechanical response of a region of a subjectbone to an applied force comprising the steps of: (a) identifying asubject bone; (b) selecting at least one database comprising empiricaldata representative of the subject bone, wherein the empirical data isbased on recorded observations of at least one cadaveric bonecorresponding to the subject bone, wherein recording the observationscomprises: (1) selecting a three-dimensional macrostructural sample fromthe cadaveric bone in which microstructural elements are present; (2)identifying at least one individual microstructural element in theselected macrostructural sample; and (3) generating viscoelastic datacomprising at least one mechanical property of the identifiedmicrostructural element obtained by subjecting the identifiedmicrostructural element to at least one force; (c) using the databasewith a computer program to generate a simulated region of the subjectbone, the simulated region comprising at least one simulatedmicrostructural element, wherein the computer program performs the stepsof: (1) defining a simulated three-dimensional region corresponding tothe simulated microstructural element of the simulated region of thesubject bone; (2) creating a finite element mesh defining a finitenumber of three-dimensional elements filling the simulatedthree-dimensional region; and (3) assigning at least one property toeach of a plurality of the elements in the finite element mesh usingviscoelastic data from the database; (d) using a computer program topredict a mechanical response of the region of the subject bone to anapplied force, wherein the computer program performs the steps of: (1)generating a simulated force applied to the simulated three-dimensionalregion; (2) calculating a response to the simulated force for each of aplurality of elements in the finite element mesh using the at least oneassigned property for each element; (3) computing a response of thesimulated three-dimensional region to the simulated force using thecalculated responses of the plurality of elements in the finite elementmesh; (4) computing the mechanical response of the simulated region ofthe subject bone from the computed response of the simulatedthree-dimensional region to the simulated force; and (5) outputting thecomputed mechanical response of the region of the subject bone to thesimulated force.
 2. The method of claim 1, wherein the database furthercomprises at least one of empirically collected dimensional data anddistributional data from at least one ultrastructural component from thesample, and elements in the finite element mesh are defined at least inpart using the ultrastructural data.
 3. The method of claim 2, whereinthe ultrastructural component is at least one of a lacuna and acanalicula from the sample.
 4. The method of claim 2, wherein elementsin the finite element mesh are representative of an ultrastructuralcomponent.
 5. The method of claim 4, wherein the ultrastructuralcomponent is at least one of a lacuna and a canalicula from the sample.6. The method of claim 1, wherein the region of the subject bonecorresponds to a lamella.
 7. The method of claim 1, wherein the regionof the subject bone corresponds to an osteon.
 8. The method of claim 1,wherein the region of subject bone corresponds to a macrostructuralregion comprising a plurality of micro structural elements.
 9. Themethod of claim 1, wherein at least one force is selected from tension,compression, shear, bending, and torsion.
 10. The method of claim 1,wherein the microstructural elements identified from the sample arelamellae.
 11. The method of claim 10, wherein the mechanical propertiesare selected from tension, compression, shear, bending, torsion,prestress, pinching, cement line slippage, stress distribution, andstrain distribution.
 12. The method of claim 10, wherein the lamellaeare one or both of bright and dark lamellae.
 13. The method of claim 1,wherein the microstructural elements identified from the sample areosteons.
 14. The method of claim 13, wherein the osteons are one or bothof longitudinal and alternate osteons.
 15. The method of claim 13,wherein the osteon mechanical properties are selected fromangle-of-twist as a function of torque per unit force, strain rate, andtime.
 16. The method of claim 1, wherein the viscoelastic data furthercomprises empirically observed orientations of collagen bundles withinmicro structural elements, and the orientation data is used in assigningproperties to elements.
 17. The method of claim 16, wherein theorientation data corresponds to lamellae organized within an osteonhaving an osteon axis, and wherein at least one dark lamella within theosteon is oriented at an angle ranging from 0 degrees to substantiallyless than ±45 degrees with respect to the osteon axis.
 18. The method ofclaim 16, wherein the orientation data corresponds to lamellae organizedwithin an osteon having an osteon axis, and wherein at least one brightlamella within the osteon is oriented at an angle ranging from amidpoint orientation with respect to the osteon axis to an angle ofabout ±45 degrees with respect to the midpoint orientation.
 19. Themethod of claim 16, wherein the orientation data corresponds to lamellaeorganized within an osteon having an osteon axis, and wherein at leastone dark lamella within the osteon is oriented at an angle of up to ±23degrees with respect to the osteon axis.
 20. The method of claim 16,wherein the orientation data corresponds to lamellae organized within anosteon having an osteon axis, and wherein at least one bright lamellawithin the osteon is oriented at an angle ranging from a midpointorientation with respect to the osteon axis to an angle of about ±23degrees with respect to the midpoint orientation.
 21. The method ofclaim 1, wherein the simulated three-dimensional region corresponding toa selected region of the subject bone is determined from an in vivo scanof the subject bone.
 22. The method of claim 8, wherein the databasefurther comprises empirical data recording the spatial distribution ofmicrostructural elements within a cadaveric sample, and the step ofgenerating a simulated region of the subject bone further comprisesdefining a spatial configuration of simulated microstructural elementsin the simulated region using data from the database.
 23. The method ofclaim 8, wherein the database further comprises empirical data recordingthe spatial distribution of microstructural elements within the subjectbone, and the step of generating a simulated region of the subject bonefurther comprises defining a spatial configuration of simulatedmicrostructural elements in the simulated region using data from thedatabase.
 24. The method of claim 23, wherein the spatial configurationof simulated microstructural elements is observed using an in vivo scanof the subject bone to obtain microstructure distribution in the subjectbone.
 25. The method of claim 1, wherein at least steps (d)(2) and d(3)are repeated for each of a plurality of simulated forces.
 26. The methodof claim 25, wherein the simulated force is applied incrementally. 27.The method of claim 1, wherein the mechanical response comprisesfracture initiation, wherein fracture initiation comprisesmicrocracking, debonding, breakage, void growth, or combinationsthereof.
 28. The method of claim 1, wherein the mechanical responsecomprises fracture propagation.
 29. The method of claim 1, wherein thestep of calculating a response to the simulated force accounts for aprestress distribution in the simulated region of the subject bone. 30.The method of claim 1, wherein the cadaveric bone is compact bone. 31.The method of claim 1, wherein the cadaveric bone is obtained from ahuman.
 32. The method of claim 1, further comprising the steps of:generating at least one distribution function corresponding to thedistribution of empirically observed values for a measurement recordedin the database; using the distribution function to assign values for atleast one property of the elements in the finite element mesh.
 33. Themethod of claim 1, wherein the subject bone corresponds to a boneattached to a prosthesis, and the applied force corresponds to forcesacting on the bone as a consequence of the attached prosthesis.
 34. Acomputerized bone model stored on one or more computer readable mediafor predicting a mechanical response of a region of a subject bone to anapplied force, comprising: (a) a database comprising empirical datarepresentative of a subject bone, wherein the empirical data comprisesrecorded observations of at least one microstructural element from acadaveric bone corresponding to the subject bone; and (b) a set ofcomputer readable instructions for use with the database comprising: (1)instructions for generating a simulated region of the subject bonewherein the simulated region comprises at least one simulatedmicrostructural element; (2) instructions for defining a simulatedthree-dimensional region corresponding to the simulated microstructuralelement of the simulated region of the subject bone; (3) instructionsfor creating a finite element mesh defining a finite number ofthree-dimensional elements filling the simulated three-dimensionalregion; (4) instructions for assigning at least one material property toeach of a plurality of the elements in the finite element mesh usingdata from the database; (5) instructions for applying a simulated forceto the simulated three-dimensional region; (6) instructions forcalculating a response to the simulated force for each of the pluralityof elements in the finite element mesh using the assigned at least oneproperty for each element; (7) instructions for computing a response ofthe simulated three-dimensional region to the simulated force using thecalculated responses of the plurality of elements in the finite elementmesh; (8) instructions for computing the mechanical response of thesimulated region of the subject bone from the computed response of thesimulated three-dimensional region to the simulated force; and (9)instructions for outputting the computed mechanical response of theregion of the subject bone to the simulated force.
 35. The computerizedbone model of claim 34 wherein the database further comprises at leastone of empirically collected dimensional data and distributional datafrom at least one ultrastructural component, and elements in the finiteelement mesh are defined at least in part using the ultrastructuraldata.
 36. The computerized bone model of claim 35, wherein theultrastructural component is at least one of a lacuna and a canalicula.37. The computerized bone model of claim 35, wherein elements in thefinite element mesh are representative of an ultrastructural component.38. The computerized bone model of claim 37, wherein the ultrastructuralcomponent is at least one of a lacuna and a canalicula.
 39. Thecomputerized bone model of claim 34, wherein the region of the subjectbone corresponds to a lamella.
 40. The computerized bone model of claim34, wherein the region of the subject bone corresponds to an osteon. 41.The computerized bone model of claim 34, wherein the region of subjectbone corresponds to a macrostructural region comprising a plurality ofmicrostructural elements.
 42. The computerized bone model of claim 34,wherein the microstructural elements identified from the sample arelamellae.
 43. The computerized bone model of claim 42, wherein themechanical properties are selected from tension, compression, shear,bending, torsion, prestress, pinching, cement line slippage, stressdistribution, and strain distribution.
 44. The computerized bone modelof claim 42, wherein the lamellae are one or both of bright and darklamellae.
 45. The computerized bone model of claim 34, wherein themicrostructural elements identified from the sample are osteons.
 46. Thecomputerized bone model of claim 45, wherein the osteons are one or bothof longitudinal and alternate osteons.
 47. The computerized bone modelof claim 45, wherein the osteon mechanical properties are selected fromangle-of-twist as a function of torque per unit force, strain rate, andtime.
 48. The computerized bone model of claim 34, wherein theviscoelastic data further comprises empirically observed orientations ofcollagen bundles within microstructural elements, and the orientationdata is used in assigning properties to elements.
 49. The computerizedbone model of claim 48, wherein the orientation data corresponds tolamellae organized within an osteon having an osteon axis, and whereinat least one dark lamella within the osteon is oriented at an angleranging from 0 degrees to substantially less than ±45 degrees withrespect to the osteon axis.
 50. The computerized bone model of claim 48,wherein the orientation data corresponds to lamellae organized within anosteon having an osteon axis, and wherein at least one bright lamellawithin the osteon is oriented at an angle ranging from a midpointorientation with respect to the osteon axis to an angle of about ±45degrees with respect to the midpoint orientation.
 51. The computerizedbone model of claim 48, wherein the orientation data corresponds tolamellae organized within an osteon having an osteon axis, and whereinat least one dark lamella within the osteon is oriented at an angle ofup to ±23 degrees with respect to the osteon axis.
 52. The computerizedbone model of claim 48, wherein the orientation data corresponds tolamellae organized within an osteon having an osteon axis, and whereinat least one bright lamella within the osteon is oriented at an angleranging from a midpoint orientation with respect to the osteon axis toan angle of about ±23 degrees with respect to the midpoint orientation.53. The computerized bone model of claim 34, wherein the simulatedthree-dimensional region corresponding to a selected region of thesubject bone is determined from an in vivo scan of the subject bone. 54.The computerized bone model of claim 41, wherein the database furthercomprises empirical data recording the spatial distribution ofmicrostructural elements within a cadaveric sample, and the set ofcomputer readable instructions further comprises instructions fordefining a spatial configuration of simulated microstructural elementsin the simulated region using data from the database.
 55. Thecomputerized bone model of claim 41, wherein the database furthercomprises empirical data recording the spatial distribution ofmicrostructural elements within the subject bone, and the set ofcomputer readable instructions further comprises instructions fordefining a spatial configuration of simulated microstructural elementsin the simulated region using data from the database.
 56. Thecomputerized bone model of claim 55, wherein the spatial configurationof simulated microstructural elements is observed using an in vivo scanof the subject bone to obtain microstructure distribution in the subjectbone.
 57. The computerized bone model of claim 34, wherein the cadavericbone is compact bone.
 58. The computerized bone model of claim 34,wherein the cadaveric bone is obtained from a human.
 59. Thecomputerized bone model of claim 34, wherein the set of computerreadable instructions further comprises: instructions for generating atleast one distribution function corresponding to the distribution ofempirically observed values for a measurement recorded in the database;instructions for using the distribution function to assign values for atleast one property of elements in the computerized model.